As an amateur, I honestly do not know what exactly to think about this question of mine. I know that there are arbitrarily large gaps between primes but do not know will they ever overlap with intervals of this kind, so that fact somehow goes in favor of that it could be that some intervals of this kind will not have primes inside (endpoints also count).
But as I also expect Polignac's conjecture to be true it could be that there will always be at least one prime in $[2^n,2^n+n^2]$.
I do not know do these intervals grow too slow but I think they (in a certain sense) do, so somehow a counterexample seems to probably exist, if you ask me, but, again, I am not sure.
Feel free to close if I made a mistake again and asked a question not appropriate for MO.