  A set system $\mathcal{F}$ is $t$-\textit{intersecting}, if the size of the
intersection of every pair of its elements has size at least $t$. A set system
$\mathcal{F}$ is $k$-\textit{Sperner}, if it does not contain a chain of length
$k+1$.
  Our main result is the following: Suppose that $k$ and $t$ are fixed positive
integers, where $n+t$ is even with $t\le n$ and $n$ is large enough. If
$\mathcal{F}\subseteq 2^{[n]}$ is a $t$-intersecting $k$-Sperner family, then
$|\mathcal{F}|$ has size at most the size of the sum of $k$ layers, of sizes
$(n+t)/2,\ldots, (n+t)/2+k-1$.
  This bound is best possible. The case when $n+t$ is odd remains open.
