  We study the version of the $k$-disjoint paths problem where $k$ demand pairs
$(s_1,t_1)$, $\dots$, $(s_k,t_k)$ are specified in the input and the paths in
the solution are allowed to intersect, but such that no vertex is on more than
$c$ paths. We show that on directed acyclic graphs the problem is solvable in
time $n^{O(d)}$ if we allow congestion $k-d$ for $k$ paths. Furthermore, we
show that, under a suitable complexity theoretic assumption, the problem cannot
be solved in time $f(k)n^{o(d/\log d)}$ for any computable function $f$.
