  We consider Schr\"odinger operators $H^h = (ih d+{\bf A})^* (ih d+{\bf A})$
with the periodic magnetic field ${\bf B}=d{\bf A}$ on covering spaces of
compact manifolds. Under some assumptions on $\bf B$, we prove that there are
arbitrarily large number of gaps in the spectrum of these operators in the
semiclassical limit of strong magnetic field $h\to 0$.
