  We present a detailed spectroscopic analysis of 115 helium-line (DB) and 28
cool, He-rich hydrogen-line (DA) white dwarfs based on atmosphere fits to
optical spectroscopy and photometry. We find that 63% of our DB population show
hydrogen lines, making them DBA stars. We also demonstrate the persistence of
pure DB white dwarfs with no detectable hydrogen feature at low effective
temperatures. Using state-of-the art envelope models, we next compute the total
quantity of hydrogen, $M_{\rm{H}}$, that is contained in the outer convection
zone as a function of effective temperature and atmospheric H/He ratio. We find
that some $(T_{\rm{eff}},M_{\rm{H}})$ pairs cannot physically exist as a
homogeneously mixed structure; such combination can only occur as stratified
objects of the DA spectral type. On that basis, we show that the values of
$M_{\rm{H}}$ inferred for the bulk of the DBA stars are too large and
incompatible with the convective dilution scenario. We also present evidence
that the hydrogen abundances measured in DBA and cool, helium-rich white dwarfs
cannot be globally accounted for by any kind of accretion mechanism onto a pure
DB star. We suggest that cool, He-rich DA white dwarfs are most likely created
by the convective mixing of a DA star with a thin hydrogen envelope; they are
not cooled down DBA's. We finally explore several scenarios that could account
for the presence of hydrogen in DBA stars.
