We conducted systematic observations of the H I Br$alpha$ (4.05 $mu$m) and Br$beta$ (2.63 $mu$m) lines in 52 nearby ($z<0.3$) ultraluminous infrared galaxies (ULIRGs) with AKARI. Among 33 ULIRGs wherein the lines are detected, three galaxies show anomalous Br$beta$/Br$alpha$ line ratios ($sim1.0$), which are significantly higher than those for case B (0.565). Our observations also show that ULIRGs have a tendency to exhibit higher Br$beta$/Br$alpha$ line ratios than those observed in Galactic H II regions. The high Br$beta$/Br$alpha$ line ratios cannot be explained by a combination of dust extinction and case B since dust extinction reduces the ratio. We explore possible causes for the high Br$beta$/Br$alpha$ line ratios and show that the observed ratios can be explained by a combination of an optically thick Br$alpha$ line and an optically thin Br$beta$ line. We simulated the H II regions in ULIRGs with the Cloudy code, and our results show that the high Br$beta$/Br$alpha$ line ratios can be explained by high-density conditions, wherein the Br$alpha$ line becomes optically thick. To achieve a column density large enough to make the Br$alpha$ line optically thick within a single H II region, the gas density must be as high as $nsim10^8$ $mathrm{cm}^{-3}$. We therefore propose an ensemble of H II regions, in each of which the Br$alpha$ line is optically thick, to explain the high Br$beta$/Br$alpha$ line ratio.
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