The experimentally well-known convergence of solvation entropies and enthalpies of different small hydrophobic solutes at universal temperatures seems to indicate that hydrophobic solvation is dominated by universal water features and not so much by solute specifics. The reported convergence of the denaturing entropy of a group of different proteins at roughly the same temperature as hydrophobic solutes was consequently argued to indicate that the denaturing entropy of proteins is dominated by the hydrophobic effect and used to estimate the hydrophobic contribution to protein stability. However, this appealing picture was subsequently questioned since the initially claimed universal convergence of denaturing entropies holds only for a small subset of proteins; for a larger data collection no convergence is seen. We report extensive simulation results for the solvation of small spherical solutes in explicit water with varying solute-water potentials. We show that convergence of solvation properties for solutes of different radii exists but that the convergence temperatures depend sensitively on solute-water potential features such as stiffness of the repulsive part and attraction strength, not so much on the attraction range. Accordingly, convergence of solvation properties is only expected for solutes of a homologous series that differ in the number of one species of subunits (which attests to the additivity of solvation properties) or solutes that are characterized by similar solute-water interaction potentials. In contrast, for peptides that arguably consist of multiple groups with widely disperse interactions with water, it means that thermodynamic convergence at a universal temperature cannot be expected, in general, in agreement with experimental results.