Is there any problem that is proved not independent of ZFC but the problem itself is not proved yet? Is there any problem in mathematics that is proved not independent of ZFC but the problem itself is not proved yet?

I'll take the question to mean, 
Is there any statement $S$ such that we can prove that $S$ is not independent of ZFC, but we have not yet been able to prove $S$, nor to refute $S$? 
Any statement $S$ that can be settled by a finite computation, but by a computation that no one has carried out yet (or by a computation so large that no one is able to carry it out), will do. For example, the question of whether the Ramsey number $R(5,5)$ is 43. For another example, according to the most recent correspondence I have from Sam Wagstaff, who keeps track of this kind of thing, no one has been able to fully factor $2^{1207}-1$ (and when/as/if that number is factored, some other number will take its place). 
