We analyze the standardizability of Type Ia supernovae (SNe Ia) in the near-infrared (NIR) by investigating the correlation between observed peak NIR (Y JH) absolute magnitude and postmaximum B-band decline rate [Delta m(15)(B)]. A sample of 27 low-redshift SNe Ia with well-observed NIR light curves observed by the Carnegie Supernova Project (CSP) between 2004 and 2007 is used. All 27 objects have premaximum coverage in optical bands, with a subset of 13 having premaximum NIR observations as well; coverage of the other 14 begins shortly after NIR maximum brightness. We describe the methods used to derive light-curve parameters (absolute peak magnitudes and decline rates) from both spline-and template-fitting procedures, and we confirm prior findings that fitting templates to SNe Ia light curves in the NIR is problematic due to the diversity of postmaximum behavior of objects that are characterized by similar Delta m(15)(B) values, especially at high decline rates. Nevertheless, we show that NIR light curves can be reasonably fit with a template, especially if the observations begin within 5 days after NIR maximum. SNe Ia appear to be better standardizable candles in the NIR bands than in the optical bands. For the subset of 13 objects in our data set that excludes the highly reddened and fast-declining SNe Ia and includes only those objects for which NIR observations began prior to 5 days after maximum light, we find modest (1.7 sigma) evidence for a peak-luminosity versus decline-rate relation in Y, and stronger evidence (2.8 sigma) in J and H. Using R-V values differing from the canonical value (R-V = 3.1) is shown to have little effect on the results. A Hubble diagram is presented for the NIR bands and the B band. The resulting scatter for the combined NIR bands is 0.13 mag, while the B band produces a scatter of 0.22 mag. Finally, we find evidence for a bimodal distribution in the NIR absolute magnitudes of fast-declining SNe Ia [Delta m(15)(B) > 1.7]. These data suggest that applying a correction to SNe Ia peak luminosities for decline rate is likely to be beneficial in the J and H bands to make SNe Ia more precise distance indicators, but of only marginal importance in the Y band.