  Atmospheric and solar neutrinos scattering coherently off target nuclei will
be an important background source for the next generation of direct dark matter
searches. In this work we focus on calcium tungstate as target material. For
comparison with existing works we calculate the neutrino floor indicating which
sensitivities can be reached before the neutrino background appears. In
addition, we investigate the sensitivities of future direct dark matter
searches using CRESST-II like detectors. Extending previous works we take into
account achievable energy resolutions and thresholds as well as beta and gamma
backgrounds for this investigation. We show that an exploration of WIMP-nucleon
cross sections below the neutrino floor is possible for exposures of
$\gtrsim10$ kg-years or higher. In the third part we show that a first
detection of coherent neutrino nucleus scattering of atmospheric and solar
neutrinos using the same detectors and the backgrounds is feasible for
exposures of $\gtrsim50$ kg-years.
