We investigate the dynamics of a realization of Fermi's relativistic acceleration mechanism that is a charged test particle oscillating between two reflecting plates that move stochastically. By allowing the charge to radiate energy during each collision, we find that the main features of the system are (1) due to the radiation drag the energy gained by the particle is bounded, and (2) the radiated energy represents a typical realization of an on-off intermittent process, due to numerous continuous encounters with a very small emission, interrupted by short and intense bursts of radiation. This intermittent radiative process exhibits non-Gaussian statistical features.