Balakran [Ba29] proved this holds for $k=1$ - that is, $(n+1)^2\mid \binom{2n}{n}$ for infinitely many $n$. It is a classical fact that $(n+1)\mid \binom{2n}{n}$ for all $n$ (see Catalan numbers).

Erdős, Graham, Ruzsa, and Straus observe that the method of Balakran can be further used to prove that there are infinitely many $n$ such that \[(n+k)!(n+1)! \mid (2n)!\] (in fact this holds whenever $k<c \log n$ for some small constant $c>0$).

Erdős [Er68c] proved that if $a!b!\mid n!$ then $a+b\leq n+O(\log n)$.