Near-infrared (NIR) spectra of the subluminous Type Ia supernova SN 1999by are presented that cover the time evolution from about 4 days before to 2 weeks after maximum light. Analysis of these data was accomplished through the construction of an extended set of delayed detonation (DD) models covering the entire range of normal to subluminous SNe Ia. The explosion, light curves, and time evolution of the synthetic spectra were calculated self-consistently for each model, with the only free parameters being the initial structure of the white dwarf and the description of the nuclear burning front during the explosion. From these, one model was selected for SN 1999by by matching the synthetic and observed optical light curves, principally the rapid brightness decline. DD models require a minimum amount of burning during the deflagration phase, which implies a lower limit for the 56Ni mass of about 0.1 M☉ and consequently a lower limit for the SN brightness. The models that best match the optical light curve of SN 1999by were those with a 56Ni production close to this theoretical minimum. The data are consistent with little or no interstellar reddening [E(B-V) ≤ 0.12 mag], and we derive a distance of 11 ± 2.5 Mpc for SN 1999by, in agreement with other estimates. Without any modification, the synthetic spectra from this subluminous model match reasonably well the observed IR spectra taken on 1999 May 6, 10, 16, and 24. These dates correspond roughly to -4, 0, 6, and 14 days after maximum light. Prior to maximum, the NIR spectra of SN 1999by are dominated by products of explosive carbon burning (O, Mg) and Si. Spectra taken after maximum light are dominated by products of incomplete Si burning. Unlike the behavior of normal Type Ia SNe, lines from iron-group elements begin to show up only in our last spectrum taken about 2 weeks after maximum light. The implied distribution of elements in velocity space agrees well with the DD model predictions for a subluminous SN Ia. Regardless of the explosion model, the long duration of the phases dominated by layers of explosive carbon and oxygen burning argues that SN 1999by was the explosion of a white dwarf at or near the Chandrasekhar mass. The good agreement between the observations and the models without fine-tuning a large number of free parameters suggests that DD models are a good description of at least subluminous Type Ia SNe. Pure deflagration scenarios or mergers are unlikely, and helium-triggered explosions can be ruled out. However, problems for DD models still remain, since the data seem to be at odds with recent three-dimensional models of the deflagration phase that predict significant mixing of the inner layers of the white dwarf prior to detonation. Possible solutions include the effects of rapid rotation on the propagation of nuclear flames during the explosive phase of burning or extensive burning of carbon just prior to the runaway.