  A \textsl{Sperner $k$-partition system} on a set $X$ is a set of partitions
of $X$ into $k$ classes such that the classes of the partitions form a Sperner
set system (so no class from a partition is a subset of a class from another
partition). These systems were defined by Meagher, Moura and Stevens in
\cite{MMS} who showed that if $|X| = k \ell$, then the largest Sperner
$k$-partition system has size $\frac{1}{k}\binom{|X|}{\ell}$. In this paper we
find bounds on the size of the largest Sperner $k$-partition system where $k$
does not divide the size of $X$, specifically, we give an exact bound when
$k=2$ and upper and lower bounds when $|X| = 2k+1$, $|X|=2k+2$ and $|X| =
3k-1$.
