We determine the osmotic pressure of microgel suspensions using membrane osmometry and dialysis, for microgels with different softnesses. Our measurements reveal that the osmotic pressure of solutions of both ionic and neutral microgels is determined by the free ions that leave the microgel periphery to maximize their entropy and not by the translational degrees of freedom of the microgels themselves. Furthermore, up to a given concentration it is energetically favorable for the microgels to maintain a constant volume without appreciable deswelling. The concentration where deswelling starts weakly depends on the crosslinker concentration, which affects the microgel dimension; we explain this by considering the dependence of the osmotic pressure and the microgel bulk modulus on the particle size.