We present high-quality spectra covering the K window at a resolving power of 340 for a sample of 13 ultraluminous (L(IR) greater than or similar to 10(12) L(.)) infrared-selected galaxies, and line fluxes for a comparison sample of 24 lower luminosity galaxies. The 2 mu m spectra of 10 of the ultraluminous galaxies are characterized by emission and absorption features commonly associated with stars and star formation; two others have the red power-law spectra and Br gamma line widths of Seyfert 1 galaxies; the final galaxy has strong emission from hot dust. We have found no broad-line active nuclei not already known from optical observations, despite the fact that the extinction at 2 mu m is 1/10 that at optical wavelengths; any putative Seyfert 1 nuclei must be deeply buried. Powerful continua and emission lines from H-2 and Br gamma are detected in all the ultraluminous galaxies. Comparing the H-2 1-0 S(1), Br gamma, and 2 mu m and far-infrared luminosities to those of the lower luminosity galaxies yields several major results. First, the dereddened Br gamma emission, relative to the far-infrared luminosity, is significantly depressed in the ultraluminous sample, when compared to the lower luminosity galaxies. Five of the ultraluminous galaxies have L(BR gamma)/L(IR) ratios lower than for any of the comparison objects. Second, the H-2 1-0 S(1) luminosity is proportional to the far-infrared luminosity, This suggests that the process producing the far-infrared luminosity is also responsible, directly or indirectly, for producing the excited H-2, and that the H-2 apparently comes from optically thin regions in both classes of objects. Third, eight of the 13 ultraluminous systems have lower 2 mu m/far-infrared luminosity ratios than any of the lower luminosity galaxies, and five of these are the galaxies also deficient in Br gamma. These three findings may be understood if the H-2, Br gamma, and 2 mu m continua in the ultraluminous galaxies arise from spatially distinct regions, with the continuum and Br gamma largely coming from volumes optically thick even at 2 mu m, and obscured in such a fashion that the extinctions measured using optical spectroscopy do not properly measure the true optical depths. If this is the case, then even near-infrared spectroscopy may be unable to exclude the presence of undetected powerful active galactic nuclei in the ultraluminous galaxies.