The NMR spin-lattice relaxation rate, ${T}_{1}^{\ensuremath{-}1},$ can be measured precisely by progressive saturation. This efficient technique is useful when ${T}_{1}$ is long and the NMR signal is weak. We derive the quasiequilibrium spin response to excitation in the case of a Zeeman spectrum in the presence of quadrupolar interactions. Exact solutions for the recovery of magnetization under the influence of purely magnetic fluctuations for $I=\frac{1}{2},$ $\frac{3}{2},$ and $\frac{5}{2}$ are presented. This is the general solution to a problem that has been previously solved only for the $I=\frac{1}{2}$ case. An important example for the application of this technique is ${}^{17}\mathrm{O}$ NMR in cuprate superconductors $(I=\frac{5}{2}).$ We show comparisons of the theory with the relaxation measured for high-temperature superconducting materials and the NMR-rates measured by this technique across the vortex-broadened spectrum at low temperature.