The convergence of multi-reference perturbation theory, in particular the CASPT method, is examined. For the HF and H2O molecules, multi-reference perturbation theory is carried through 20th order and the results are compared with full configuration interaction (FCI) calculations. The CASPT multi-reference perturbation theory is found to be divergent for wavefunctions with significant amounts of static correlation. However, the correlation energy estimates obtained at second and up to fourth order provide in general a sequence of improved approximations to the FCI results. The divergence of the perturbation expansion is analyzed in terms of the eigenvalue spectrum of the Hamiltonian and is shown to be caused by low-lying backdoor-intruder states.