We report a new experimental study of the growth of longitudinal energy spread in a space-charge-dominated electron beam, with a beam energy of several keV and beam current of approximately 100 mA. At relatively low beam densities, we measure growing energy spreads with distance along the transport channel, which are in remarkably good agreement with the theory of energy relaxation via Coulomb collisions. At higher beam densities, however, anomalous energy spreads exceeding the predictions of the relaxation theory are observed, which, we believe, could be caused by collective longitudinal-transverse instabilities observed in computer simulation studies. The onset of these instabilities occurs after several plasma periods according to calculations.