We consider a general system of coupled nonlinear diffusion equations that are characterized by having degenerate source terms and thereby not having isolated rest states. Using a general form of physically relevant source terms, we derive conditions that are required to trigger traveling waves when a stable uniform steady-state solution is perturbed by a highly localized disturbance. We show that the degeneracy in the source terms implies that traveling waves have a number of surprising properties that are not present for systems with nondegenerate source terms. We also show that such systems can lead to a pair of waves that initially propagate outwards from the disturbance, slow down, and reverse direction before ultimately colliding and annihilating each other.