In the temperature range 6000<Teff<8500 K, the atmospheric properties of the majority of white dwarfs observed are well reproduced by either hydrogen- or helium-rich models. However, a homogeneous group of peculiar objects—first identified by Bergeron, Ruiz, and Leggett—constitutes an important exception to this rule: while the energy distributions of these stars are better represented by pure hydrogen models, their spectra do not show the Hα line, even though that line should be visible down to ~4000 K. The existence of these objects, together with that of a non-DA gap in the range 5100<Teff<6100 K, prompted Bergeron and coworkers to propose a process by which large amounts of hydrogen can be accreted onto the helium-rich atmosphere of a cooling white dwarf, while still remaining invisible. We explore in detail the physical basis of this proposed mechanism and show that it cannot account for the pattern of atmospheric abundances observed below 8500 K. As an alternative, we consider the possibility that the current generation of cool, helium-rich atmosphere models might omit a significant opacity source. The role played by the He2 quasi molecule in these models deserves continued scrutiny.