We consider a model of dark energy interacting with dark matter. An interaction may be interesting for the solution of the coincidence problem. We model dark energy as a fluid with constant equation of state parameter w and study background and linear perturbations. While non-interacting models with constant w are well behaved, a simple interaction leads to an instability in the dark sector perturbations at early times. The instability is severe, non-adiabatic and independent of how weak the coupling is.