A family of sets is $t$-intersecting if any two sets from the family contain at least $t$ common elements. Given a $t$-intersecting family of $r$-sets from an $n$-set, how many distinct sets of size $k$ can occur as pairwise intersections of its members? We prove an asymptotic upper bound on this number that can always be achieved. This result can be seen as a generalization of the Erdős-Ko-Rado theorem.