  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 HI 21-cm transition. Here we estimate the HI gas mass of a sample
of main-sequence star-forming galaxies at $z\sim 6.5 - 7.8$ surveyed for
[CII]$-158\mu$m emission as part of the Reionization Era Bright Emission Line
Survey (REBELS), using a recent calibration of the [CII]-to-HI conversion
factor. We find that the HI gas mass excess in galaxies increases as a function
of redshift, with an average of $M_{\rm HI} / M_\star \approx 10$,
corresponding to HI gas mass fractions of $f_{\rm HI} = M_{\rm HI} / (M_\star +
M_{\rm HI}) = 90\%$, at $z\approx 7$. Based on the [CII]-$158\mu$m luminosity
function (LF) derived from the same sample of galaxies, we further place
constraints on the cosmic HI gas mass density in galaxies ($\rho_{\rm HI}$) at
this redshift, which we measure to be $\rho_{\rm HI} = 7.1^{+6.4}_{-3.0} \times
10^{6}\,M_{\odot}\,{\rm Mpc^{-3}}$. This estimate is substantially lower by a
factor of $\approx 10$ than that inferred from an extrapolation of damped
Lyman-$\alpha$ absorber (DLA) measurements, and largely depend on the exact
[CII] LF adopted. However, we find this decrease in $\rho_{\rm HI}$ to be
consistent with recent simulations and argue that this apparent discrepancy is
likely a consequence of the DLA sightlines predominantly probing the
substantial fraction of HI gas in high-$z$ galactic halos, whereas [CII] traces
the HI in the ISM associated with star formation. We make predictions for this
build-up of neutral gas in galaxies as a function of redshift, showing that at
$z\gtrsim 5$ only $\approx 10\%$ of the cosmic HI gas content is confined in
galaxies and associated with the star-forming ISM.
