Nuclear-magnetic-resonance data are presented and analyzed for the high-${\mathit{T}}_{\mathit{c}}$ compound ${\mathrm{Bi}}_{2}$${\mathrm{Sr}}_{2}$${\mathrm{CaCu}}_{2}$${\mathrm{O}}_{8}$ for two oxygen doping levels. Both sample conditions lead to spin-gap behavior for the NMR shift, with a precursive downturn in the data at Tg${\mathit{T}}_{\mathit{c}}$. In addition, the relaxation times ${\mathit{T}}_{1}$ obey the relation (${\mathit{T}}_{1}$T${)}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\propto}${\mathit{K}}^{\mathit{s}}$(T) at low temperatures (T\ensuremath{\lesssim}100 K), where ${\mathit{K}}^{\mathit{s}}$(T) is the spin paramagnetic shift. This relation, which is also obeyed by other superconductors, is argued to be related to the spin-gap effects and thus incompatible with a Fermi-liquid approach to the understanding of these systems.