We used the Space Telescope Imaging Spectrograph (STIS) with its smallest entrance aperture (003 wide slit) and highest resolution echelle gratings (E140H and E230H) to record the interstellar absorption features for 10 different multiplets of neutral carbon at a resolving power of λ/Δλ = 200,000 in the UV spectra of 21 early-type stars. Our objective was to measure the amount of C I in each of its three fine-structure levels of the ground electronic state, so that we could determine the thermal pressures in the absorbing gas and how much they vary in different regions. Our observations are principally along directions out to several kiloparsecs in the Galactic plane near longitudes l = 120^° and 300°, with the more distant stars penetrating nearby portions of the Perseus and Sagittarius-Carina arms of the Galaxy. We devised a special analysis technique to decipher the overlapping absorption features in the different multiplets, each with different arrangements of the closely spaced transitions. In order to derive internally consistent results for all multiplets, we found that we had to modify the relative transition f-values in a way that made generally weak transitions stronger than amounts indicated in the current literature. We compared our measured relative populations of the excited fine-structure levels to those expected from equilibria calculated with collisional rate constants for various densities, temperatures, and compositions. The median thermal pressure for our entire sample was p/k = 2240 cm^-3 K, or slightly higher if the representative temperatures of the material are much above or below a most favorable temperature of 40 K for the excitation of the first excited level at a given pressure. For gas that is moving outside the range of radial velocities permitted by differential Galactic rotation between us and the targets, about 15% of the C I indicates a thermal pressure p/k > 5000 cm^-3 K. For gas within the allowed velocities, this fraction is only 1.5%. This contrast reveals a relationship between pressure enhancements and the kinematics of the gas. Regardless of velocity, we usually can register the presence of a very small proportion of the gas that seems to be at p/k 10⁵ cm^-3 K. We interpret these ubiquitous wisps of high-pressure material to arise either from small-scale density enhancements created by converging flows in a turbulent medium or from warm turbulent boundary layers on the surfaces of dense clouds moving through an intercloud medium. For turbulent compression, our C I excitations indicate that the barytropic index γ_eff 0.90 must apply if the unperturbed gas starts out with representative densities and temperatures n = 10 cm^-3 and T = 100 K. This value for γ_eff is larger than that expected for interstellar material that remains in thermal equilibrium after it is compressed from the same initial n and T. However, if regions of enhanced pressure reach a size smaller than ~0.01 pc, the dynamical time starts to become shorter than the cooling time, and γ_eff should start to approach the adiabatic value c_p/c_v = 5/3. Some of the excited C I may arise from the target stars' H II regions or by the effects of optical pumping from the submillimeter line radiation emitted by them. We argue that these contributions are small, and our comparisons of the velocities of strongly excited C I to those of excited Si II seem to support this outlook. For six stars in the survey, absorption features from interstellar excited O I could be detected at velocities slightly shifted from the persistent features of telluric origin. These O I* and O I** features were especially strong in the spectra of HD 93843 and HD 210839, the same stars that show exceptionally large C I excitations. In appendices of this paper, we present evidence that (1) the wavelength resolving power of STIS in the E14OH mode is indeed about 200,000, and (2) the telluric O I* and O I** features exhibit some evidence for macroscopic motions, since their broadenings are in excess of that expected for thermal Doppler broadening at an exospheric temperature T = 1000 K.