Dissolved paramagnetic ions generally provide an efficient mechanism for the relaxation of nuclear spins in solution, a phenomenon called the nuclear magnetic resonance-paramagnetic relaxation enhancement (NMR-PRE). Metal ions with electron spins S greater than or equal to 1 exhibit rich NMR relaxation phenomena originating in the properties of the zero-field splitting (zfs) interaction, which vanishes for spin-1/2 ions but which is nonzero for S greater than or equal to 1 ions in site symmetry lower than cubic. For S greater than or equal to 1 ions in the vicinity of the zfs-limit, i.e., at magnetic-field strengths low enough that the zfs energy exceeds the Zeeman energy, the NMR-PRE depends strongly on the detailed structure of the electron spin energy levels as well as on the spatial quantization of the spin motion. It is shown theoretically and experimentally that the NMR-PRE produced by integer spins can be influenced strongly by the small intradoublet zero-field splittings, i.e., the splittings between the components of the non-Kramers doublets, which are produced by noncylindrical components of the crystal field potential. These small splittings produce relatively low-frequency oscillations in the dipolar field associated with [S-(z over cap)] (the spin component along the molecule-fixed (z) over cap axis). These motions decouple the nuclear spin from the electron spin, thereby depressing, in some cases very strongly, the NMR-PRE. The presence of a relatively small Zeeman field, comparable in magnitude to the intradoublet spacing but small compared to the larger interdoublet zfs splittings, causes a major change in the spin wave functions which has profound effects on the motions of the electron spin. When the Zeeman energy exceeds the small zfs splitting, the oscillatory motion of [S-(z over cap)] damps out, with the result that the electron spin couples more effectively to the nuclear spin, providing a more efficient NMR relaxation pathway. NMR-PRE data are presented for the S = 1 complex Ni(II)(o-pda)(2)Cl-2 (o-pda = ortho-phenylenediamine) which confirm the importance of the splitting of the m(S) = +/-l non-Kramers doublet on the NMR relaxation efficiency. The zfs E-parameter was measured from the NMR data to be \E\ = 0.26 cm(-1). The S = 2 spin system, Mn(lll) -tetraphenylporphyrin sulfonate, exhibits a related phenomenon which arises from the effects of a small zfs splitting, Delta epsilon(+/-2), of the m(S) = +/-2 non-Kramers doublet that is caused by a fourfold rotational component of the crystal field potential. The splitting Delta epsilon(+/-2) was measured from NMR data to be 0.20 cm(-1).