We consider the problem of distributed compression for correlated quantum sources. The classical version of this problem was solved by Slepian and Wolf, who showed that distributed compression could take full advantage of redundancy in the local sources created by the presence of correlations. We show that, in general, this is not the case for quantum sources by proving a lower bound on the rate sum for irreducible sources of product states which is stronger than the one given by a naive application of Slepian-Wolf. Nonetheless, strategies taking advantage of correlation do exist for some special classes of quantum sources. For example, Devetak and Winter demonstrated the existence of such a strategy when one of the sources is classical. Here we find optimal non-trivial strategies for a different extreme, sources of Bell states. In addition, we illustrate how distributed compression is connected to other problems in quantum information theory, including information-disturbance questions, entanglement distillation and quantum error correction.