Whitehead problem is a rather well known problem:

Suppose that $G$ is an abelian group and $\mathrm{Ext}^1(G,\Bbb Z)=0$, is $G$ free?

It wasn't long before it was proved that if $G$ is countable (and thus countably-generated) then the answer is positive. However the question was open for uncountable groups, and it took more than a decade until Shelah came up with the interesting answer: *In ZFC the problem is undecidable*.

I was wondering what other open problems have somewhat of a Whitehead-like nature, namely they were proved for objects with a countable nature (e.g. separable topological spaces, countably generated), but the question remains open for the general case.

One example which comes to mind is Naimark's problem which was solved for separable $C^\ast$-algebras, and was partially solved in the general case in the sense that it is consistent to have a negative answer.

(As with CW big-lists, please post one problem per answer.)