Depending if 2014 or 2024 rules, it may not be possible because the revision tp Spellcasting now include this:
One Spell with a Spell Slot per Turn
On a turn, you can expend only one spell slot to cast a spell. This rule means you can’t, for example, cast a spell with a spell slot using the Magic action and another one using a Bonus Action on the same turn.
Situation
Mage X: cast a spell, disintegration as exemple
PG mage: use reaction and cast counterspell vs Mage X spell
Mage X: use reaction and cast counterspell vs PG counterspell and PG fail TS.
Mage X continue with disintegration spell
Its right?
Depending if 2014 or 2024 rules, it may not be possible because the revision tp Spellcasting now include this:
As @Plaguescarred said, not possible under the 2024 rules.
If more examples would help, I recommend this post: Reaction spells - Rules & Game Mechanics
Well, it is possible if either the original spell or the counter-counter are cast without a spell slot (item some kind of special ability)
But no, in the general case it's not. You'll need to have two mages in the party to back each other up. (I have this as a DM. It's so annoying. :)