Much of the interstellar medium in disk galaxies is in the form of neutral atomic hydrogen, H I. This gas can be in thermal equilibrium at relatively low temperatures, T<~300 K (the cold neutral medium [CNM]), or at temperatures somewhat less than 104 K (the warm neutral medium [WNM]). These two phases can coexist over a narrow range of pressures, Pmin<=P<=Pmax. We determine Pmin and Pmax in the plane of the Galaxy as a function of Galactocentric radius R using recent determinations of the gas heating rate and the gas-phase abundances of interstellar gas. We provide an analytic approximation for Pmin as a function of metallicity, far-ultraviolet radiation field, and the ionization rate of atomic hydrogen. Our analytic results show that the existence of Pmin, or the possibility of a two-phase equilibrium, generally requires that H+ exceed C+ in abundance at Pmin. The abundance of H+ is set by EUV/soft X-ray photoionization and by recombination with negatively charged polycyclic aromatic hydrocarbons. In order to assess whether thermal or pressure equilibrium is a realistic assumption, we define a parameter Υ≡tcool/tshock, where tcool is the gas cooling time and tshock is the characteristic shock time or ``time between shocks in a turbulent medium.'' For Υ<1 gas has time to reach thermal balance between supernova-induced shocks. We find that this condition is satisfied in the Galactic disk, and thus the two-phase description of the interstellar H I is approximately valid even in the presence of interstellar turbulence. Observationally, the mean density HI> is often better determined than the local density, and we cast our results in terms of HI> as well. Over most of the disk of the Galaxy, the H I must be in two phases: the weight of the H I in the gravitational potential of the Galaxy is large enough to generate thermal pressures exceeding Pmin, so that turbulent pressure fluctuations can produce cold gas that is thermally stable; and the mean density of the H I is too low for the gas to be all CNM. Our models predict the presence of CNM gas to R~=16-18 kpc, somewhat farther than previous estimates. An estimate for the typical thermal pressure in the Galactic plane for 3kpc<~R<~18 kpc is Pth/k~=1.4×104exp(-R/5.5kpc) K cm-3. At the solar circle, this gives Pth/k~=3000 K cm-3. We show that this pressure is consistent with the C I*/C Itot ratio observed by Jenkins & Tripp and the CNM temperature found by Heiles & Troland. We also examine the potential impact of turbulent heating on our results and provide parameterized expressions for the heating rate as a function of Galactic radius. Although the uncertainties are large, our models predict that including turbulent heating does not significantly change our results and that thermal pressures remain above Pmin to R~=18 kpc.