We reexamine the question of the relationship between the onset of pulsational driving and the thickness of the outer hydrogen layer in ZZ Ceti stars. To this end, we solve the system of ordinary differential equations which describes the linear, nonadiabatic, nonradial pulsations of stellar models with the help of a new code based on the powerful Galerkin finite-element method of weighted residuals. We find that the theoretical blue edge temperature of the ZZ Ceti instability strip is insensitive to the hydrogen layer mass for log (M(H)/M*) greater-than-or-equal-to - 12. This is in line with mounting evidence that ZZ Ceti pulsators with relatively thick hydrogen layers do actually exist, but it is contrary to some previous calculations which have been widely accepted. We discuss the implications of our findings vis-a-vis current white dwarf spectral evolution models.