The profiles of H alpha in a number of cool Ap stars are anomalous. Broad w
ings, indicative of temperatures in the range 7000-8000 K end abruptly in n
arrow cores. The widths of these cores are compatible with those of dwarfs
with temperatures of 6000 K or lower. This profile has been known for Przyb
ylski's star, but it is: seen in other cool Ap stars. The H beta profile in
several of these stars shows a similar core-wing anomaly (CWA). In Przybyl
ski's star, the CWA is probably present at higher Balmer members. We are un
able to account for these profiles within the context of LTE and normal dwa
rf atmospheres. We conclude that the atmospheres of these stars are not "no
rmal". This is contrary to a notion that has long been held.