The convergence of the Moller-Plesset expansion is examined for Ne, F-, CH2, and HF and analyzed by means of a simple two-state model. For all systems, increasing diffuseness of the basis introduces highly excited diffuse back-door intruder states, resulting in an an alternating, ultimately divergent expansion. For F-, the divergence begins already at third order; for the remaining systems, it begins later. For CH2, the low-lying doubly excited state leads to a monotonic, slowly decreasing series at lower orders; for the stretched HF molecule, the low-lying doubly excited states lead to a slowly undulating series at lower orders. Although the divergence of the Moller-Plesset series does not invalidate the use of the second-order expansion, it questions the use of higher-order Moller-Plesset expansions in quantum-chemical studies.