The accepted explanation of the observed dichotomy of two orders of magnitude between in situ measurements of electron density in H II regions, derived from emission line ratios, and average measurements based on integrated emission measure, is the inhomogeneity of the ionized medium. This is expressed as a ``filling factor", the volume ratio of dense to tenuous gas, measured with values of order 10-3. Implicit in the filling factor model as normally used, is the assumption that the clumps of dense gas are optically thin to ionizing radiation. Here we explore implications of assuming the contrary: that the clumps are optically thick. A first consequence is the presence within H II regions of a major fraction of neutral hydrogen. We estimate the mean Ho/H+ ratio for a population of H II regions in the spiral galaxy NGC 1530 to be the order of 10, and support this inference using dynamical arguments. The optically thick clumpy models allow a significant fraction of the photons generated by the ionizing stars to escape from their H II region. We show, by comparing model predictions with observations, that these models give an account at least as good as, and probably better than that of conventional models, of the radial surface brightness distribution and of selected spectral line diagnostics for physical conditions within H II regions. These models explain how an H SHAPE Ii region can appear, from its line ratios, to be ionization bounded, yet permit a major fraction of its ionizing photons to escape.