We present new Green Bank Telescope (GBT) 21 cm neutral hydrogen (H I) observations of a complete distance-limited sample of 22 Hickson Compact Groups (HCGs) with at least four true members. We detected an average H I mass of 8 × 109 M (median = 6 × 109 M ), which is significantly larger than previous single-dish measurements. Consequently, the H I deficiencies for these HCGs have been reduced, although not completely eliminated. Spectral comparison of the GBT data with complementary Very Large Array data shows significant H I excess in the GBT spectra. The observed excess is primarily due to the high surface brightness (HSB) sensitivity of the GBT detecting diffuse, low column density H I in these groups. The excess gas forms a faint diffused neutral medium which is an intermediate stage in the evolution of HSB H I tidal debris in the intragroup medium (IGM) before it is fully ionized. The excess gas mass fraction, (M(H I)GBT – M(H I)VLA)/M(H I)GBT, for our complete sample varies from 5% to 81% with an average of 36% (median = 30%). The excess gas mass fraction is highest in slightly H I deficient groups where the tidal debris has had enough time to evolve. We also find the excess gas content increases with the evolutionary phase of the group described in Verdes-Montenegro et al. Theoretical calculations indicate that an H I cloud of radius ≥ 200 pc would survive in an IGM of 2 × 106 K for more than the typical dynamical lifetime of a group. However, smaller clouds get evaporated and assimilated into the hot IGM in a much shorter timescale.