The potential application of carbon nanotubes as electron sources in electron microscopes is analyzed. The resolution and probe current that can be obtained from a carbon nanotube emitter in a low-voltage scanning electron microscope are calculated and compared to the state of the art using Schottky electron sources. Many analytical equations for probe-size versus probe-current relations in different parameter regimes are obtained. It is shown that for most carbon nanotube emitters, the gun lens aberrations are larger than the emitters' virtual source size and thus restrict the microscope's performance. The result is that the advantages of the higher brightness of nanotube emitters are limited unless the angular emission current is increased over present day values or the gun lens aberrations are decreased. For some nanotubes with a closed cap, it is known that the emitted electron beam is coherent over the full emission cone. We argue that for such emitters the parameter "brightness" becomes meaningless. The influence of phase variations in the electron wave front emitted from such a nanotube emitter on the focusing of the electron beam is analyzed.