The convergence of Moller-Plesset perturbation expansions (MP2-MP4/MP5) for the spectroscopic constants of a selected set of diatomic molecules (BH, CH, HF, N-2, CO, and F-2) has been investigated. It was found that the second-order perturbation contributions to the spectroscopic constants are strongly dependent on basis set, more so for HF and CO than for BH. The MP5 contributions for HF were essentially zero for the cc-pVDZ basis set, but increased significantly with basis set illustrating the difficulty of using small basis sets as benchmarks for correlated calculations. The convergence behavior of the exact Moller-Plesset perturbation expansions were investigated using estimates of the complete basis set limits obtained using large correlation consistent basis sets. For BH and CH, the perturbation expansions of the spectroscopic constants converge monotonically toward the experimental values, while for HF, N-2, CO, and F-2, the expansions oscillate about the experimental values. The perturbation expansions are, in general, only slowly converging and, for HF, N-2, CO, and F-2, appear to be far from convergence at MP4. In fact, for HF, N-2, and CO, the errors in the calculated spectroscopic constants for the MP4 method are larger than those for the MP2 method (the only exception is D-e). The current study, combined with other recent studies, raises serious doubts about the use of Moller-Plesset perturbation theory to describe electron correlation effects in atomic and molecular calculations.