High brightness ultrafast electron sources are limited by space-charge effects. In the high-electron-density regime, Coulomb repulsion induces both transverse and longitudinal broadenings of electron packets. We describe a three-dimensional mean-field model, which takes into account Coulomb repulsion in all directions. Measurements of spatial electron pulse widths are carried out after extraction from a photoemission electron gun. Experiments and simulation results are in very good agreement. Coulomb repulsion effects are found to occur close to the photocathode in the extraction region. The correlation between transverse and temporal broadenings of electron packets is discussed and gives the limits for the generation of high-electron-density ultrafast electron pulses. Pulses of temporal length of about 2 ps and a diameter of about 30 μm should be obtained.