The number of spells you can know ("known spells") is different from how many you can cast ("spell slots"). As a 9th level bard, you have 12 known spells (as shown in the bard class table). So out of those 12 known spells, you can cast 4 level 1 spells, 3 level 2 spells, 3 level 3 spells, 3 level 4 spells, and 1 level 5 spell before having to take a long rest. Those 12 known spells can be of any level you have spell slots for, and you pick them from the Bard spell list.
Multiclassing spellcasters does make things more complicated (although everything is pretty clearly explained in the Multiclassing section of the Basic Rules). However, since you are multiclassing bard (which has the Spellcasting feature) and warlock (which has the similar but different Pact Magic feature), it pretty much works like a 9th level bard's spell slots plus a 4th level warlock's pact magic slots.
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I have a 9th level bard/4th level warlock (archfey)
According to the spell progression chart, I should have 3 4th level spells and 1 5th level spell.
However, I can only select 1 4th level and 1 5th level spell. After that, all the other Learn button grey out.
Am I missing something? Did I do it wrong? Is the warlock class interfering? Or is it like a ability limiting the amount of spells knowable?
Here's the character: https://www.dndbeyond.com/characters/59033262/6bEYQV
EDIT: I figured it out. I had four 1st level spells. Once I moved to only knowing three 1st level spells, it was fine. LOL.
The number of spells you can know ("known spells") is different from how many you can cast ("spell slots").
As a 9th level bard, you have 12 known spells (as shown in the bard class table). So out of those 12 known spells, you can cast 4 level 1 spells, 3 level 2 spells, 3 level 3 spells, 3 level 4 spells, and 1 level 5 spell before having to take a long rest.
Those 12 known spells can be of any level you have spell slots for, and you pick them from the Bard spell list.
Multiclassing spellcasters does make things more complicated (although everything is pretty clearly explained in the Multiclassing section of the Basic Rules). However, since you are multiclassing bard (which has the Spellcasting feature) and warlock (which has the similar but different Pact Magic feature), it pretty much works like a 9th level bard's spell slots plus a 4th level warlock's pact magic slots.