Recent studies have seriously questioned the use of higher-order Moller-Plesset perturbation theory (MPn) in describing electron correlation in atomic and molecular systems. Here we first reinvestigate with improved numerical techniques previously controversial and disturbing MPn energetic series for Ne, F-, HF, BH, C-2 and N-2. Conspicuously absent in previous work is research on the convergence of MPn spectroscopic constants, and thus complete MPn (energy, r(e), omega(e)) series were then computed for (BH, HF, CN+, C-2 and N-2) through the high orders (MP25, MP21, MP13, MP39 and MP19) within the correlation consistent family of basis sets. A persistent, slowly decaying ringing pattern in the C-2 energy series was tracked out to MP155. Finally, new energy series were generated and analyzed through MP167 for Cl- and MP39 for Ar and HCl. The MPn energy and property series variously display rapid or slow convergence, monotonic or oscillatory decay, highly erratic or regular behavior, or early or late divergence, all depending on the chemical system or the choice of one-particle basis set. For oscillatory series the spectroscopic constants computed from low-order MPn methods are often more accurate, with respect to the full configuration interaction (FCI) limit, than those computed via high-order MPn theory.