The dependence of the spin-pumping effect on the yttrium iron garnet (Y3Fe5O12, YIG) thickness detected by the inverse spin Hall effect (ISHE) has been investigated quantitatively. Due to the spin-pumping effect driven by the magnetization precession in the ferrimagnetic insulator Y3Fe5O12 film a spin-polarized electron current is injected into the Pt layer. This spin current is transformed into electrical charge current by means of the ISHE. An increase of the ISHE voltage with increasing film thickness is observed and compared to the theoretically expected behavior. The effective damping parameter of the YIG/Pt samples is found to be enhanced with decreasing Y3Fe5O12 film thickness. The investigated samples exhibit a spin mixing conductance of g↑↓eff=(3.87±0.21)×1018m−2 and a spin Hall angle between θISHE=0.013±0.001 and 0.045±0.004 depending on the used spin-diffusion length. Furthermore, the influence of nonlinear effects on the generated voltage and on the Gilbert damping parameter at high excitation powers is revealed. It is shown that for small YIG film thicknesses a broadening of the linewidth due to nonlinear effects at high excitation powers is suppressed because of a lack of nonlinear multimagnon scattering channels. We have found that the variation of the spin-pumping efficiency for thick YIG samples exhibiting pronounced nonlinear effects is much smaller than the nonlinear enhancement of the damping.