  The accuracy of the conventional treatment of quark mass thresholds in the
QCD running coupling constant $\alpha_s$, based on the step approximation to
the $\beta$--function, is investigated. The errors of extrapolating $\ase(\mu)$
from low energies to $\mu=M_Z$, implied by this approximation, are shown to be
of the same magnitude as the typical next--to--next--to--leading order
contributions to \as and tend to increase the resulting \as. The importance of
the proper choice of matching points is emphasized.
