In this paper we describe the effect of imperfections in the radio environment map (REM) information on the performance of cognitive radio (CR) systems. Via simulations we explore the relationship between the required precision of the REM and various channel/system properties. For example, the degree of spatial correlation in the shadow fading is a key factor as is the interference constraint employed by the primary user. Based on the CR interferers obtained from the simulations, we characterize the temporal behavior of such systems by computing the level crossing rates (LCRs) of the cumulative interference represented by these CRs. This evaluates the effect of short term fluctuations above acceptable interference levels due to the fast fading. We derive analytical formulae for the LCRs in Rayleigh and Rician fast fading conditions. The analytical results are verified by Monte Carlo simulations.