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 YIG 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 YIG film thickness. The investigated samples exhibit a spin mixing conductance of g=(7.43 ± 0.36) \times 10^{18} m^{-2} and a spin Hall angle of theta_{ISHE} = 0.009 ± 0.0008. Furthermore, the influence of nonlinear effects on the generated voltage and on the Gilbert damping parameter at high excitation powers are 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 multi-magnon 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.