High-dispersion IUE spectra of Al III, Si IV, C IV, and N V for 12 sight lines in the Galactic halo are presented. The spectra for each sight line are co-added combinations of three to seven individual IUE images and have S/N approximately 10-25. We find that the Si IV, C IV, and N V absorption profiles are considerably different from the profiles of Al III, a tracer of photoionized gas in the interstellar medium. We detect N v, a tracer of hot ( T approximately 2 x 10(5) K) collisionally ionized gas along 10 of the 12 sight lines. The observed Si IV and C IV profiles correlate more closely with one another than with the Al III profiles, indicating that a significant fraction of the gas along these sight lines may be collisionally ionized. Evidence for photoionized Si IV and C IV along a few sight lines is presented. The observed widths of the absorption lines increase with successively higher ionization states. The line-width differences are larger than those expected from thermal line broadening of gas at the temperatures where these ions peak in abundance, implying that there may be large-scale differences in the distribution of the Al III-, Si IV-, C IV-, and N V-bearing gas. The observed column density ratios of the high ions are: [ N(C IV)/N(Si IV) ] = 3.6 +/-1.3; [N(C IV)/N(N V) ] = 4.6 +/- 2.7. We find that these ratios are remarkably constant throughout the Galaxy and do not depend upon the distance of the absorbing gas away from the Galactic plane. The observed column densities and ratios are discussed within the context of current theoretical models of the production of highly ionized gas in the Galactic halo. None of the photoionization or collisional ionization models can simultaneously reproduce the observed column densities and ratios of Si IV, C IV, and N V. Photoionization models which incorporate an extragalactic radiation field predict a large increase in high-ionization atom density near absolute value of z approximately 1 kpc. Such an increase is not observed. Recent models incorporating self-ionization of cooling hot gas and various hybrid photoionization/Galactic fountain models may be necessary to reproduce the measurements.