  We discuss physical conditions in Galactic neutral hydrogen based on deep,
high velocity resolution interferometric HI 21 cm absorption spectroscopy
towards 33 compact extra-galactic radio sources. The HI 21 cm optical depth
spectra have root-mean-square noise values \lesssim 10^{-3} per 1 km/s velocity
channel, i.e., sufficiently sensitive to detect HI 21 cm absorption by the warm
neutral medium (WNM). Comparing these spectra with HI 21 cm emission spectra
from the Leiden-Argentine-Bonn (LAB) survey, we show that some of the
absorption detected on most sightlines must arise in gas with temperatures
higher than that in the stable cold neutral medium (CNM). A multi-Gaussian
decomposition of 30 of the HI 21 cm absorption spectra yielded very few
components with line widths in the temperature range of stable WNM, with no
such WNM components detected for sixteen of the thirty sightlines. We find that
some of the detected HI 21 cm absorption along thirteen of these sightlines
must arise in gas with spin temperatures larger than the CNM range. For these
sightlines, we use very conservative estimates of the CNM spin temperature and
the non-thermal broadening to derive strict upper limits to the gas column
densities in the CNM and WNM phases. Comparing these upper limits to the total
HI column density, we find that typically at least 28% of the gas must have
temperatures in the thermally unstable range (200-5000 K). Our observations
hence robustly indicate that a significant fraction of the gas in the Galactic
interstellar medium has temperatures outside the ranges expected for thermally
stable gas in two-phase models.
