This is a variant of Is there a statement whose undecidability is undecidable? and Can it be shown that ZFC has statements which cannot be proven to be independent, but are? (but is not asked or answered in either of those threads).
Call a formula $\phi$ provably undecidable if $ZFC$ proves that ($ZFC$ is consistent) $\Rightarrow$ ($\phi$ is independent of ZFC). Now, (assuming that $ZFC$ is consistent) are there statements which are undecidable but not provably so ?
Here is another way to put it. Call a statement $\phi$ weakly decidable if ZFC can either prove that $\phi$ is true or prove that $\phi$ is false, or at least prove the implication ($ZFC$ is consistent) $\Rightarrow$ ($\phi$ is independent of ZFC).
Thus, the continuum hypothesis is weakly decidable in ZFC, and the axiom of choice is weakly decidable in ZF. On the other hand, it is not known today if the existence of an inacessible cardinal is weakly decidable.
The original question can then be rephrased as, are some statements not even weakly decidable ?