We present evidence that the H II regions of high luminosity in disk galaxies may be density bounded, so that a significant fraction of the ionizing photons emitted by their exciting OB stars escapes from the regions. The key piece of evidence is the presence of glitches, local sharp peaks at an apparently invariant luminosity, in the H α luminosity functions of the populations of H II regions. The apparently invariant luminosity is defined as the Strömgren luminosity (L Str), such that L H α= L Str= 10 38.6±10 0.1 ergs s-1 (no other peaks are found in any of the luminosity functions) accompanying a steepening of slope for L H α> L Str. This behavior is readily explicable by a physical model in which (1) the transition at L H α= L Str marks a change from essentially ionization bounding at low luminosities to density bounding at higher values and (2) for this to occur the law relating stellar mass in massive star-forming clouds to the mass of the placental cloud must be such that the ionizing photon flux produced within the cloud is a function that rises more steeply than the mass of the cloud. Supporting evidence for the hypothesis of this transition is also presented: measurements of the central surface brightnesses of H II regions for L H α< L Str are proportional to L, as expected for ionization bounding, but show a sharp trend to a steeper dependence for L H α> L Str, and the observed relation between the internal turbulence velocity parameter, σ, and the luminosity, L, at high luminosities can be well explained if these regions are density bounded. If confirmed, the density-bounding hypothesis would have a number of interesting implications. It would imply that the density-bounded regions were the main sources of the photons that ionize the diffuse gas in disk galaxies. Our estimates, based on the hypothesis, indicate that these regions emit sufficient Lyman continuum photons not only to ionize the diffuse medium but to cause a typical spiral to emit significant ionizing flux into the intergalactic medium. The low scatter observed in L Str, less than 0.1 mag rms in the still quite small sample measured to date, is an invitation to widen the database and to calibrate against primary standards to obtain a precise (∼ 10 5 L⊙) widely distributed standard candle.