We present a comparison of experimentally and theoretically determined osmotic pressures for various colloidal dispersions. Experimental data is collected from several different silica and polystyrene dispersions. The theoretical pressure determinations are based on the primitive model combined with the cell model, and the physical quantities are calculated exactly using Monte Carlo simulations in the canonical and grand canonical ensemble. The input to the simulations in terms of colloidal particle size, surface charge density, and so forth are taken directly, from experiments, and the approach does not contain any adjustable parameters. The agreement between theory and experiment is very good without any fitting parameters, showing that the simplifications behind the primitive model and the cell model are physically sound. The results reveal a surprising correspondence between the equations of state in spherical and planar geometries, indicating that the particle shape is of secondary importance in dispersions dominated by repulsive interactions. For one of the silica dispersions, we have also investigated how various monovalent counterions influence the swelling properties. Within experimental error, we are unable to detect any ion specificity, which is further support for the theoretical models used.