Erdős and Graham write that $n+1$ always divides $\binom{2n}{n}$ (indeed $\frac{1}{n+1}\binom{2n}{n}$ is the $n$th Catalan number), but it is quite rare that $n$ divides $\binom{2n}{n}$.

Pomerance [Po14] has shown that for any $k\geq 0$ there are infinitely many $n$ such that $n-k\mid\binom{2n}{n}$, although the set of such $n$ has upper density $<1/3$. Pomerance also shows that the set of $n$ such that \[\prod_{1\leq i\leq k}(n+i)\mid \binom{2n}{n}\] has density $1$.

The smallest $n$ for each $k$ are listed as A375077 on the OEIS.