DB and DBA white dwarfs are generally believed to be the result of a process by which a thin radiative hydrogen atmosphere floating in diffusive equilibrium on top of a helium envelope is eventually completely diluted in the underlying more massive helium convection zone that develops with cooling. However, the observed hydrogen abundances in these objects exceed by several orders of magnitude the predictions obtained from such a scenario invoking diffusive equilibrium, thus currently leaving the very existence of DB and DBA white dwarfs unaccounted for in any satisfactory way. We present here the results of new numerical simulations aimed at improving the modeling of this convective dilution process. In particular, we show how DA white dwarfs can be transformed into DB stars below 20,000 K, and more importantly, we propose a model that predicts the correct amount of hydrogen observed in DBA stars without invoking any accretion mechanism, an alternative model that has been proposed over the years to account qualitatively for the presence of hydrogen in the atmospheres of DBA stars.