Is there always at least one prime in these closed intervals?
The first counterexample:
For $n = 13$, $2^n = 8192, 2^n + n = 8205$.
There does not exist any prime in $[8192,8205]$, though $8191$ is a prime.
The second counterexample:
For $n =14, 2^n = 16384, 2^n + n = 16398$.
There does not exist any prime in $[16384,16398]$.
Checked with this list.
Wikipedia references a theorems that put a lower bound on the size of gaps you can expect. For any positive constant $C$, there exists infinitely many integers $n$ such that the interval
$$ (n, n + C \log n) $$
does not contain any primes at all.
You're asking about gaps of size $\log_2 n$; consequently, your conjecture can be summed up as
The powers of 2 are very rare; maybe we get lucky and they never land near the beginning of such a wide gap?
Still, one might ask if your conjecture is plausible. Heuristically, assuming primes are distributed "randomly", the odds that an interval $(n, n + \log_2 n)$ doesn't contain a prime is roughly
$$ \left( 1 - \frac{1}{\log n} \right)^{\log_2 n } \approx \exp\left(-\frac{1}{\log 2} \right) \approx 0.2363$$
as $n$ grows large. Consequently, you'd expect roughly one in every four values of $n$ to serve as a counterexample.
Even if you amend your conjecture to account for finitely many counterexamples, it is implausible without some rationale suggesting there's a causal reason the intervals you've chosen should be special in some fashion.