  Detailed Monte Carlo simulations of the expected radioactive background rates
and spectra in NaI crystals are presented. The obtained spectra are then
compared to those measured in the DAMA/NaI and DAMA/LIBRA experiments. The
simulations can be made consistent with the measured DAMA spectrum only by
assuming higher than reported concentrations of some isotopes and even so leave
very little room for the dark matter signal. We conclude that any
interpretation of the annual modulation of the event rate observed by DAMA as a
dark matter signal, should include full consideration of the background
spectrum. This would significantly restrict the range of dark matter models
capable of explaining the modulation effect.
