  A quantitative study of the field-induced magnetic ordering in TlCuCl$_3$ in
terms of a Bose-Einstein condensation (BEC) of magnons is presented. It is
shown that the hitherto proposed simple BEC scenario is in quantitative and
qualitative disagreement with experiment. It is further shown that even very
small Dzyaloshinsky-Moriya interactions or a staggered $g$ tensor component of
a certain type can change the BEC picture qualitatively. Such terms lead to a
nonzero condensate density for all temperatures and a gapped quasiparticle
spectrum. Including this type of interaction allows us to obtain good agreement
with experimental data.
