In this article, we critically study whether social networks can explain the emergence of cooperative behavior. We carry out an extensive simulation program in which we study the most representative social dilemmas. For the Prisoner's Dilemma, it turns out that the emergence of cooperation is dependent on the microdynamics. On the other hand, network clustering mostly facilitates global cooperation in the Stag Hunt game, whereas degree heterogeneity promotes cooperation in Snowdrift dilemmas. Thus, social networks do not promote cooperation in general, because the macro-outcome is not robust under change of dynamics. Therefore, having specific applications of interest in mind is crucial to include the appropriate microdetails in a good model.