We present both high-resolution (R = 1260) and low-resolution (R = 345 and 425) J-band spectra of a sample of 15 Seyfert 2 galaxies. Our goal is to look for broad Pabeta lines, indicating broad-line regions which are hidden by dust from our view at optical wavelengths. Our long-term goal is to use Pabeta and other IR lines to determine the reddenings along the lines of sight to the hidden BLRs. With this information for a statistically large sample, we can then recreate the profile of the dust optical depth as a function of polar angle in the obscuring torus. Of the 15 objects studied here, three have broad Pabeta lines: MCG-05.23.16, Mrk 463E, and NGC 2992. Mrk 176 and NGC 5728 may also have weak broad lines; somewhat improved data should prove definitive. In NGC 5506, previously reported to have broad Pabeta and Halpha lines, we find that the Pabeta line profile is continuous and has the same shape as the nearby line [Fe II] lambda1.2567, which should not have a broad component. We interpret these observations as gas from the narrow-line region (NLR) with no broad component. In NGC 5506, however, the NLR profiles become broader with increasing wavelength, indicating that highly reddened wings are becoming more readily visible at the longer wavelengths. Our wavelength range also includes [Fe II] lambda1.2567, thought to come either from supemova remnants in a starburst or from high optical depth NLR gas heated by X-rays. We confirm the correlation of [O I] lambda6300/Halpha and [Fe II] lambda1.644/Brgamma (the latter transformed to [Fe II] lambda1.2567/Pabeta to compare with our data) found by previous authors when comparing active galactic nuclei (AGNs), supernova remnants, starbursts, and H II regions. The correlation confirms that in all of these objects both [O I] A6300 and the [Fe II] lines come from partially ionized regions in which hydrogen is mostly neutral. Comparison of the infrared optical depths with column depths determined from X-ray data show a general tendency for the objects with detected broad Pabeta to have lower X-ray columns, although this should be confirmed with IR spectroscopy of other lines and better X-ray spectra.