Nonlinear microresonators are very desired for a wide variety of applications. Up-conversion processes responsible for the transformation of IR laser radiation into visible are intensity-dependent and thus rather sensitive to all involved effects, which can mask each other. In this work we study the phenomena that are the most important for possible lasing in 4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4 H-pyran dye spherical microresonators: the two-photon absorption and photobleaching. Based on the suggested model of the threshold-like dependence of the two-photon luminescence (TPL) on pump power, we demonstrate the role of intensity-dependent photobleaching in the appearance of the TPL and find a good agreement with the experiment. This finding is important for the analysis of lasing in nonlinear dye-based resonators.