  We present a study of the evolution of the column density distribution,
f(N,z), and total neutral hydrogen mass in high-column density quasar absorbers
using candidates from a recent high-redshift survey for damped Lyman-alpha
(DLA) and Lyman limit system (LLS) absorbers. The observed number of LLS
(N(HI)> 1.6 * 10^{17} atom/cm^2) is used to constrain f(N,z) below the
classical DLA Wolfe et al. (1986) definition of 2 * 10^{20} atom/cm^2. The
joint LLS-DLA analysis shows unambiguously that f(N,z) deviates significantly
from a single power law and that a Gamma-law distribution of the form
f(N,z)=(f_*/N_*)(N/N_*)^{-Beta} exp(-N/N_*) provides a better description of
the observations. These results are used to determine the amount of neutral gas
contained in DLAs and in systems with lower column density. Whilst in the
redshift range 2 to 3.5, ~90% of the neutral HI mass is in DLAs, we find that
at z>3.5 this fraction drops to only 55% and that the remaining 'missing' mass
fraction of the neutral gas lies in sub-DLAs with N(HI) 10^{19} - 2 * 10^{20}
atom/cm^2. The characteristic column density, N_*, changes from 1.6 * 10^{21}
atom/cm^2 at z<3.5 to 2.9 * 10^{20} atom/cm^2 at z>3.5, supporting a picture
where at z>3.5, we are directly observing the formation of high column density
neutral hydrogen DLA systems from lower column density units. Moreover since
current metallicity studies of DLA systems focus on the higher column density
systems they may be giving a biased or incomplete view of global galactic
chemical evolution at z>3. After correcting the observed mass in HI for the
``missing'' neutral gas the comoving mass density now shows no evidence for a
decrease above z=2. (abridged)
