The only times it fits are as if the character is completely out of practice (which I personally don't like the idea) or having an injury. 

Do you think that most characters in low levels are supposed to be quite new in the field or inexperienced in it?

Ah yes, this is also known as the "old wizard" problem. Why would the old wizard who has spent years studying magic only know cantrips and a couple 1st level spells? This is where creative backstories can come into play. I had one where a wish spell caused them to have memory loss so they had to relearn magic. Being out of practice (while boring) is a legit thing. Maybe a soldier took up farming after a war and has not picked up a weapon since. What LilithFairen posted is right on. Just come up with a reason that fits your character and works for you as a backstory and you're good. 

Agree with both the above and their examples on how to make those characters work.

To add yet another option, class changes are a good reason to start a character with a long backstory at level one.  A person who has been a fighter all their life, but takes an oath to become a paladin might have to learn their skills from scratch to accommodate the different fighting styles inherent in their new class.  They might still have some residual skills - that’s what backgrounds are for (and this might get even better and more able to show residual skills when the improved backgrounds of 5.5 are released).

Caerwyn’s idea of multiclassing works ( more or less) in 5e but worked far better back in 1e where humans couldn’t multiclass but could switch class - with the caveat that while they kept their HP s and skills they couldn’t use any of the features of their previous class until they were a level higher in the new than the old at which point they could use all the features of both classes. While it’s certainly not RAW  I could see a DM a version of this to multiclassing with the added effect of making 1 level dips less enticing.

Level 1 Characters are experienced with every day tasks, problems and the dangers connected with their profession, they might even have fought in wars and survived, hunted dangerous beasts and succeeded. A veteran to anyone who just learned at which end to hold their sword/wand/arrow,  but confronted with extraordinary situations like  interdimensional invasions or ancient dragons burning a whole country, (or even just a flock of owlbears in the woods) even a veteran will know they either run or die.

The difference is those individuals having the aspiration to do research, train and acquire the necessary gear to deal with these kind of problems, moving on from veterans to masters and specialists, as there probably have been others before them. Heroes (or villains) of legend. And as there is strength in numbers, they gang up with equals on similar paths.

I do think that characters starting at low levels should be young for their race. Sometimes very young. And very inexperienced in ALL areas.

Such as if a character has a skill like smithing from a backstory it should be a very limited skill set.
Such as only able reform simple metal items. Like nails and at most horse shoes. Single part items. Not things like hinges or buckles. Rivets maybe. They can resharpen weapons but not weld parts back together. They can shorten a broken long sword into a short sword but reweld the broken off tip to keep it a long sword. They can knock out dents but not fix holes. They can polish metals to a clean surface but not polish them to a mirror like shine.
All of this work can be done in the field with a fire, simple bellows, hammer, tong, file, sharpening stone and a good rock for an anvil. Less than 10 pounds of gear total.

Any back story skill should require a set amount of tools that a can be carried normally and weigh as little as possible. Think of it as a hobby. Simple leather working tools could fit into a pouch. The same with a good set of lock picks. or even simple forgery tools.

A low-level character is raw, but there’s still a lot of room to work with; a soldier might have had a long term of low-intensity service, making them a veteran without having pushed them to develop their skills beyond the basics. Same deal for any caster; even if they’ve been out practicing their craft for a while, if they’ve only been in low-intensity situations they won’t necessarily been forced to improve their skills the way an active PC is.

Granted, the other side of this is also that if a player is looking for realism in backstory vs level, they do need to adapt that backstory to accommodate the campaign’s starting point sometimes. A first tier character isn’t going to have a collection of feats and achievements making them famed in song and story.

Quote from Caerwyn_Glyndwr >>

Agree with both the above and their examples on how to make those characters work.

To add yet another option, class changes are a good reason to start a character with a long backstory at level one.  A person who has been a fighter all their life, but takes an oath to become a paladin might have to learn their skills from scratch to accommodate the different fighting styles inherent in their new class.  They might still have some residual skills - that’s what backgrounds are for (and this might get even better and more able to show residual skills when the improved backgrounds of 5.5 are released).

I do this "playing against type" background a lot -- druids with the entertainer background, warlocks with the soldier background, that sort of thing. Adventurers can have entire lives before hitting level 1

The trope of the young unskilled beginner that becomes a great hero/artisan/etc is a well used one - mostly in teen/“young adult”/kid focused stories as the readers are similar to the protagonists. It can work in D&D at least for some classes. One of the best is Robert Asprin’s Myth Inc. series with a young Klahd kid trying to become a mage and having learned 2 spells (levitate and create flame) facing ridiculous situations and coming through with the help of friends and the fact that he is a good kid that just keeps trying to do his best. But the reality is that for tool sets and music instruments proficiency is not “well I can maybe do something”  it’s real actual skill. In the Middle Ages that meant being an adult (well 16+ anyway) with a real skill - not an apprentice but a journeyman (or more). While playing against type can make for a more interesting role play experience for some classes (like ranger) the skills may seem minor but are anything but and any individual that hasn’t spent significant time learning them is doomed (or should be). Some classes - like sorceror and warlock - are easy to do against type  while other classes - like ranger, Paladin, artificer etc don’t really work well as they call for real skills that take time to be learned and honed.

Quote from Skfma >>

The only times it fits are as if the character is completely out of practice (which I personally don't like the idea) or having an injury. 

Do you think that most characters in low levels are supposed to be quite new in the field or inexperienced in it?

If you work with XP, which you get by killing things, if you have been posted as a guard at a fort in a peaceful part of the kingdom you can easily have years of experience soldiering without actually having gotten (or forgotten) anything that would make you advance from level 1.

Same with milestones...

The vast majority of everyone everywhere is a class-less schmuck with abilities around 10. This includes nobility, royalty, and the rank and file of every army of every race. The city guard isn't a multi-class level 12 - no, he's a guy with a cudgel. The tavern keeper isn't a retired adventurer with a magical sword above the mantle - no, he's a guy who serves beer for money. The king's trusted advisor isn't a level 20 wizard - but he's actually a classless schmuck with Int or Wis or both scores of above 10. As is the king - possibly. His position is, after all, inherited, and thus falls to the eldest son, no matter the qualifications. 

And thus, a veteran can be a guy with simply better ability scores. Or a girl with some class levels. 

But in my games I try very hard to make it clear that random people they meet in the street - including soldiers, and people in robes or carrying holy symbols - have no class.

Acromos'

That is not my idea of NPC's or common people.
Common solders are obviously a low level with their rank as a relative reflection of their experience and level. And obvious;y someone needs to teach them so those people should have a bit more experience.
Common clerics need to be able to at least cast a few low level spells.

As for rulers and those other people. This is something that D&D has always been short of, non adventuring classes like diplomats and rulers. Experience used to be given for a successful encounter, not just the defeat of an opponent. So as the diplomat makes successful deals he gains experience and thus gains levels.
This can also be expressed as a merchant class or farmer class. The vast majority of them my never get over level 3 in their class but they do gain experience.

Quote from TheGnome5786 >>

Acromos'

That is not my idea of NPC's or common people.
Common solders are obviously a low level with their rank as a relative reflection of their experience and level. And obvious;y someone needs to teach them so those people should have a bit more experience.
Common clerics need to be able to at least cast a few low level spells.

As for rulers and those other people. This is something that D&D has always been short of, non adventuring classes like diplomats and rulers. Experience used to be given for a successful encounter, not just the defeat of an opponent. So as the diplomat makes successful deals he gains experience and thus gains levels.
This can also be expressed as a merchant class or farmer class. The vast majority of them my never get over level 3 in their class but they do gain experience.

 Well, we don't have to agree =)

A common cleric needs to be able to preach. A common soldier needs to be able to march, hold a spear and a shield, and die - preferably after inflicting at least some damage on the enemy.

Rulers needn't have any class. Being born a duke doesn't imbue you with anything at all except a title. 

In my game world, most everyone never even get to level 1. They are, quite literally, classless for the entirety of their lives. And that goes for commoners as well as for nobility.

We know it's possible for a commoner to kill, say, a giant. Giants only have attacks, no half-damage-on-a-sucessful-saving-throw effects, so a giant could roll 1s for all its attacks and the commoner could roll nothing but 20s, and the giant would lose.

That would take an incredible amount of luck, but if it happened, you can just about guarantee that the commoner who did it is destined for greatness.

To me, that's an adventurer. Adventuring is a ridiculous gamble, and you get better at playing the odds with practice. An adventurer isn't necessarily better, in my estimation, at ordinary things, than an ordinary person. But ordinary things aren't high-stakes enough to necessitate using a d20 to resolve them. In a properly considered game, they're not using Sleight of Hand to pick pockets for a few dollars, they're using it to steal diamonds from a dragon. They're not swinging swords on the front line of an army, they're doing it underwater while being grappled by a giant octopus. These aptitudes stem from common backgrounds, yes, but when you roll a skill, I don't think the question the d20 asks is "how good are you at this skill." That's what the proficiency bonus is for. That's a hard and fast fact. The die asks, "how well can your experience with this skill pull you through this situation?"

But y'know. It's so easy just to do the other thing. Roll for everything, low rolls mean you're suddenly inept, everybody's clamoring for a skill points variant rule so you can play Skyrim while you play D&D. I get it. I think it's wrong and bad, but I do get the appeal. And in those games, you need to have higher level NPCs and lower level PCs, you need a smaller and more granular range of aptitudes. And in those games it doesn't make sense for you to be able to start as a veteran at level 1 unless you essentially say you're not "really" that experienced because you forgot or went into a coma or something. Sure.

Quote from ChoirOfFire >>

We know it's possible for a commoner to kill, say, a giant. Giants only have attacks, no half-damage-on-a-sucessful-saving-throw effects, so a giant could roll 1s for all its attacks and the commoner could roll nothing but 20s, and the giant would lose.

 Literally David vs. Goliath.

Granted, ostensibly divine intervention was involved in that particular scenario, which underlines why it's a bit out there as a low level backstory.

So I have actually done this for a PbP campaign I'm in. My character is a Shadow Magic Sorcerer. He began as a 14th level sorc adventurer but he was betrayed by his party who was hired by his father (although the father may not know it was actually his son [which he believed was dead], and just needed the power of a shadow-infused sorc). They siphoned his magical power which also weakened his body and mind. After this on his return to find his adoptive mother sick, she died and he fell into a depressive state for years - burying himself in vices, squandering away his wealthy estate and such indulgences have impacted his health and mind as well. He eventually came out of it all but due to everything he was incredibly weakened in his magic, in his body, his skills and his mind. 

This helps develop his reason for joining the adventure - to seek out leads of his former group to get answers and revenge. However, there is no expectation of this : the former group, his father, and all of that are just there if the DM wants to make use of them but I most certainly don't expect any such things. It's just a way to motivate him and explain why he is the way he is.

Although he was a "former adventurer" I will be certain to avoid metagaming. I'm not going to be "oh I've faced this monster before, what we need to do is.." because that's tacky. Generally most things will be "new things" which can be explained either by being genuinely new or him not remembering things properly due to everything that affected his mind. If I think there's a chance he might know something I would check with the DM who can say yes, no or make a check or something. I haven't done this yet and if I do, I will note it to ensure I don't do so again for quite some time. 

So, yeah, it's a fun way to explore a backstory, justify background choices, and explain character motivations and behaviours. But, you do need to be careful to ensure you're no better than any other 1st level character and to avoid any metagaming.

Maybe they once were mighty, but almost died and must regain their strength, breath of the wild style.