I recall (though it might be my faulty memory) an exercise in some book (perhaps Jech?) that was along the lines
"Suppose $\mathrm{GCH}$ holds in $V$ and $P$ is a Cohen forcing that adds $\kappa$ subsets to $\lambda$, then in $V[G]$ there exists some $\beta$ such that $\mathrm{GCH}$ holds above $\aleph_\beta$".
Of course there is an assumption that $P$ is a set, otherwise Easton's theorem tells us otherwise.
If this is indeed from Jech's Set Theory then it is most likely correct, and I cannot see the reason why; if it is a result of some bad sectors in my brain, I still cannot come up with a counterexample.
(Also, if it is from some book I would be glad to have the reference!)