Is there any way for improved familiars w / class levels to gain more levels?

[Reduxist]

I was looking up the list for the Improved Familiar feat's options and I saw that a few of the options such as the Tidepool Dragon and the Faerie Dragon have levels in spellcasting classes such as Sorceror. Are there any ways to give them more levels in those classes or are they stuck at those levels?

[Tiny Coffee Golem]

Not without DM Fiat.

Edit: though I think Beast Bomded witch can assign her feats to her familiar. That's kinda close.

[avr]

Release them from being a familiar & take the leadership feat?

[Mufflebear]

The thing is, there is nothing that says you can't. And more importantly, there is an Improved familiar in wrath of the righteous that has an impressive 17 class levels divided into 3 different classes. This suggests that you can indeed have a familiar with levels.

[Hushed]

If you take them as an Improved Familiar, not that I know of.

If you take them as a Guardian Spirit yes... maybe? I haven't been able to get a clear answer. It may have to be adjudicated by your GM.

[Quandary]

And FYI, the familiar creatures you mention don't HAVE SORCEROR LEVELS.
They CAST AS a Sorceror of X levels, i.e. spellslots, spellsknown, CL.
But they don't even have a Sorceror Bloodline never mind other features.
If they do take further Sorceror levels, they start with Level 1 Sorceror
as far as class abilities, but spellcasting stacks above this.
Most dragons have similar spellcasting, which also stacks w/ class levels.

There is no mechanical way for Feat/Class-granted Familiar to do this.
But as a Monster/NPC it is reasonable to allow adding class levels,
and these would certainly count as "related class" or whatever for CR.

[Azothath]

A familiar is a class "feature". So they are part of the class and scale accordingly. They are part of the class and linked to class level.

PCs and some independent NPCs gain experience. Essentially being active NPCs they would probably be cohorts gained via Leadership. In the game we assume BBEGs and other NPCs got their levels by gaining experience. What? You thought GMs just made them up??? lol... where do you think experience from TPKs goes to? ;-)
So you'd need something more independent than a familiar or animal companion.
Lastly, your GM *could* figure something out, but that's up to him.

=====
when a pair of adventurers under one player is done like this, usually it's through Leadership and usually the fantastical creature is the prime and the less fantastical one the cohort. This is done as the fantastical creature usually requires more levels.
I had a pair; Jimminy & Crixit. Jimminy was a high level rouge and Crixit a pixie Wizard. Of course Crixit was the prime and had leadership although Jimminy seemed to be the leader.
In Living Arcanis Organized Play the Patrician(politician) was the prime and the Wizard the cohort (seen in play). This is because the Patrician could have a cohort Patrician+1 levels (yes, 1 level above the character with Leadership).