Many combinatorial problems are NP-complete for general graphs, but are not NP-complete for partial k-trees (graphs of treewidth bounded by a constant k) and can be efficiently solved in polynomial time or mostly in linear time for partial k-trees. On the other hand, very few problems are known to be NP-complete for partial k-trees with bounded k. These include the subgraph isomorphism problem and the bandwidth problem. However, all these problems are NP-complete even for ordinary trees or forests. In this paper we show that the edge-disjoint paths problem is NP-complete for partial k-trees with some bounded k, say k = 3, although the problem is trivially solvable for trees.