  Let $\N_n$ be the set of nilpotent $n$ by $n$ matrices over an algebraically
closed field $k$. For each $r\ge 2$, let $C_r(\N_n)$ be the variety consisting
of all pairwise commuting $r$-tuples of nilpotent matrices. It is well-kown
that $C_2(\N_n)$ is irreducible for every $n$. We study in this note the
reducibility of $C_r(\N_n)$ for various values of $n$ and $r$. In particular it
will be shown that the reducibility of $C_r(\mathfrak{gl}_n)$, the variety of
commuting $r$-tuples of $n$ by $n$ matrices, implies that of $C_r(\N_n)$ under
certain condition. Then we prove that $C_r(\N_n)$ is reducible for all $n, r\ge
4$. The ingredients of this result are also useful for getting a new lower
bound of the dimensions of $C_r(\N_n)$ and $C_r(\mathfrak{gl}_n)$. Finally, we
investigate values of $n$ for which the variety $C_3(\N_n)$ of nilpotent
commuting triples is reducible.
