  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 $n\sim10^8$
$\mathrm{cm}^{-3}$. 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.
