  Let $S$ be a smooth projective surface over $\mathbb{C}$. Let
$S^{[n_1,\dots,n_k]}$ denote the nested Hilbert scheme which parametrizes
zero-dimensional subschemes $\xi_{n_1} \subset \ldots \subset \xi_{n_k}$ where
$\xi_i$ is a closed subscheme of $S$ of length $i$. We show that $S^{[n,m]}$,
$S^{[n,m,m+1]}$, $S^{[n,n+1,m]}$, $S^{[n,n+1,m,m+1]}$, $S^{[n,n+2,m]}$ and
$S^{[n,n+2,m,m+1]}$ are irreducible.
