A detailed analysis of the strengths and weaknesses of coupled-cluster and many-body perturbation theories in treating strongly interacting potential energy surfaces is presented. Standard coupled cluster theory is shown to provide a qualitative treatment of Herzberg-Teller coupling that is vastly superior to that associated with perturbation theory. However, it also predicts unphysical effects that will always cause it to fail in describing the topology of potential energy surfaces in the immediate vicinity of conical intersections. To treat problems involving strong interstate coupling (notably those involving radicals subject to pseudo-Jahn-Teller effects), methods based on equation-of-motion (linear response) coupled-cluster theory appear to be considerably more suitable. In particular, they provide a description of intersecting surfaces that is qualitatively correct in all respects. It is also shown that there is no reason to believe that the noniterative inclusion of triple excitation contributions to the correlation energy should provide for any systematic improvement in describing this class of phenomena.