Cononsolvency refers to the effect in which a polymer chain in good solvent collapses when it is mixed with an increasing concentration of a second good solvent. An important question is if universality of nonspecific attraction between monomers and cosolvent explains the cononsolvency phenomenon. We report results obtained with computer simulations of coarse-grained models and show that the above universality does not exist. We demonstrate that cononsolvency can occur with or without the preferential adsorption of the cosolvent on the polymer chain and discuss experimental systems corresponding to these cases.