The CRESST experiment monitors 300g CaWO_4 crystals as targets for particle interactions in an ultra low background environment. In this paper, we analyze the background spectra that are recorded by three detectors over many weeks of data taking. Understanding these spectra is mandatory if one wants to further reduce the background level, and allows us to cross-check the calibration of the detectors. We identify a variety of sources, such as intrinsic contaminations due to primordial radioisotopes and cosmogenic activation of the target material. In particular, we detect a 3.6keV X-ray line from the decay of 41-Ca with an activity of (26pm4)mu Bq, corresponding to a ratio 41-Ca/40-Ca=(2.2pm0.3)times10^{-16}.
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