The growth of cell colonies is determined by the migration and proliferation of the individual cells. This is often modeled with the Fisher-Kolmogorov (FK) equation, which assumes that cells diffuse independently from each other, but stop to proliferate when their density reaches a critial limit. However, when using measured, cell-line specific parameters, we find that the FK equation drastically underestimates the experimentally observed increase of colony radius with time. Moreover, cells in real colonies migrate radially outward with superdiffusive trajectories, in contrast to the assumption of random diffusion. We demonstrate that both dicrepancies can be resolved by assuming that cells in dense colonies are driven apart by repulsive, pressure-like forces. Using this model of proliferating repelling particles (PRP), we find that colony growth exhibits different dynamical regimes, depending on the ratio between a pressure-related equilibrium cell density and the critial density of proliferation arrest.