We investigate the feasibility of bremsstrahlung radiation from "nonthermal" electrons as a source of hard X-rays from the intracluster medium of clusters of galaxies. With an exact treatment of the Coulomb collisions in a Fokker-Planck analysis of the electron distribution, we find that the severe difficulties with lifetimes of nonthermal particles found earlier by Petrosian in 2001 using a cold-target model remain problematic. We then address the possible acceleration of background electrons into a nonthermal tail. We assume a simplified but generic acceleration rate and determine the expected evolution of an initially Maxwellian distribution of electrons. We find that strong nonthermal components arise only for a rapid rate of acceleration, which also heats up the entire plasma. These results confirm the conclusion that if the observed nonthermal excesses are due to some process accelerating the background thermal electrons, then this process must be short lived.
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