The neutral atomic gas content of individual galaxies at large cosmological distances has until recently been difficult to measure due to the weakness of the hyperfine H I 21 cm transition. Here we estimate the H I gas mass of a sample of main-sequence star-forming galaxies at z similar to 6.5-7.8 surveyed for [C II] 158 mu m emission as part of the Reionization Era Bright Emission Line Survey (REBELS), using a recent calibration of the [C II]-to-H I conversion factor. We find that the H I gas mass excess in galaxies increases as a function of redshift, with an average of M-HI /M-star approximate to 10, corresponding to H I gas mass fractions of f(HI) = M-HI /(M-star + M-HI) = 90%, at z approximate to 7. Based on the [C II] 158 mu m luminosity function (LF) derived from the same sample of galaxies, we further place constraints on the cosmic H I gas mass density in galaxies (rho(HI)) at this redshift, which we measure to be rho(HI) = 7.1(-3.0)(+6.4) x 10(6) M-circle dot Mpc(-3). This estimate is substantially lower by a factor of approximate to 10 than that inferred from an extrapolation of damped Ly alpha absorber (DLA) measurements and largely depends on the exact [C II] LF adopted. However, we find this decrease in rho(HI) to be consistent with recent simulations and argue that this apparent discrepancy is likely a consequence of the DLA sight lines predominantly probing the substantial fraction of H I gas in high-z galactic halos, whereas [C II] traces the H I in the ISM associated with star formation. We make predictions for this buildup of neutral gas in galaxies as a function of redshift, showing that at z greater than or similar to 5, only approximate to 10% of the cosmic H I gas content is confined in galaxies and associated with the star-forming ISM.