Near-infrared spectroscopy (lambda(rest)similar to 3700-6800 Angstrom) of eight high-redshift powerful radio galaxies (HzPRGs) at z = 2.2-2.6 is presented. Strong forbidden lines and Her emission were detected in all sources; the data show evidence that the emission lines of the HzPRGs may contribute a substantial fraction (similar to 25%-98%) of their total observed H- and/or K-band light. Diagnostic emission-line ratios-[O III] lambda 5007/K beta versus [S II] lambda lambda 6716, 6731/H alpha-for three of the eight HzPRGs are consistent with the presence of a Seyfert 2 nucleus; the [O III] lambda 5007/H beta and [S II] lambda lambda 6716, 6731/H alpha ratios and/or limits of the remaining five galaxies are inconclusive. Furthermore, all six of the galaxies for which both H-and K-band spectra were obtained have observed [O III] lambda 5007/(H alpha + [N II] lambda lambda 6548, 6583) ratios consistent with Seyfert 2 ionization. Much of the inability to detect the weaker emission lines of [S II] lambda lambda 6716, 6731 in three of the galaxies and HE in any of the galaxies may be caused by moderate amounts of dust: for the two sources with previously measured Ly alpha fluxes, the observed Ly alpha/H alpha ratios are similar to 1.5, much less than the value of 16 expected for gas in a dust-free medium photoionized by a hard, nonthermal continuum. If such a discrepancy is caused solely by dust, this ratio translates into A(v) similar to 0.5-1.0 mag (depending on which extinction curve-Milky Way, SMC, or LMC-is used) at the rest frame optical wavelengths of the galaxies and a corresponding factor of similar to 1.6-2.5 reduction in optical flux. None of the eight HzPRGs at z = 2.2-2.6 has a broad (Delta v(FWHM) > 1500 km s(-1)) emission-line core, and it is not clear whether any have broad emission-line wings. However, the near-infrared spectrum of 3C 22, a z = 0.937 radio galaxy with 1 mu m luminosity comparable to that of the radio galaxies at z = 2.2-2.6 and a radio luminosity only 3-5 times less, shows direct evidence for broad Ha! emission wings. Such a feature is indicative of the presence of a partially obscured Seyfert 1 nucleus. Given that 3C 22 is at similar to 1/3 the luminosity distance of the sample of HzPRGs at z = 2.2-2.6, a thorough search for such a faint feature in the more distant galaxies may require 8-10 m class telescopes. These new data, along with recent UV-to-optical polarimetry showing evidence of high polarization in many HzPRGs, provide evidence that many HzPRGs are predominantly ionized by an active nucleus, and that a significant fraction of their spectral energy distribution may be caused by nonthermal emission from an active galactic nucleus (AGN).