We have obtained 1087 NIR (JHK$_s$) measurements of 21 SNe Ia using PAIRITEL, nearly doubling the number of well-sampled NIR SN Ia light curves. These data strengthen the evidence that SNe Ia are excellent standard candles in the NIR, even without correction for optical light-curve shape. We construct fiducial NIR templates for normal SNe Ia from our sample, excluding only the three known peculiar SNe Ia: SN 2005bl, SN 2005hk, and SN 2005ke. The H-band absolute magnitudes in this sample of 18 SNe Ia have an intrinsic rms of only 0.15 mag with no correction for light-curve shape. We found a relationship between the H-band extinction and optical color excess of A$_H$ = 0.2E(B - V) . This variation is as small as the scatter in distance modulus measurements currently used for cosmology based on optical light curves after corrections for light-curve shape. Combining the homogeneous PAIRITEL measurements with 23 SNe Ia from the literature, these 41 SNe Ia have standard H-band magnitudes with an rms scatter of 0.16 mag. The good match of our sample with the literature sample suggests there are few systematic problems with the photometry. We present a nearby NIR Hubble diagram that shows no correlation of the residuals from the Hubble line with light-curve properties. Future samples that account for optical and NIR light-curve shapes, absorption, spectroscopic variation, or host-galaxy properties may reveal effective ways to improve the use of SNe Ia as distance indicators. Since systematic errors due to dust absorption in optical bands remain the leading difficulty in the cosmological use of supernovae, the good behavior of SN Ia NIR light curves and their relative insensitivity to reddening make these objects attractive candidates for future cosmological work.