  It is well known that when the distance $t$ traveled by neutrinos in matter
is short, matter effects in oscillations of neutrino flavour are small and
decrease with decreasing $t$ more rapidly than the oscillations effects
themselves. We discuss the reason for this and demonstrate that under certain
circumstances this statement is no longer correct. In particular, we show that
if neutrinos propagate significant distances in vacuum before entering matter
(or after exiting it), matter effects in short-$t$ neutrino oscillations can be
significantly enhanced. Implications for oscillations of solar and atmospheric
neutrinos with nearly horizontal trajectories inside the earth and for
accelerator experiments are considered. We also comment on neutrino
oscillations in matter due to flavour changing neutral currents.
