We present early-time infrared (IR) and optical spectroscopy, and optical photometry, of the Type Ia supernova 1994D. These observations provide the most complete optical-IR spectral coverage ever achieved for a Type Ia at this phase. Optical and IR spectra were obtained as early as 9d before maximum light. The combined optical and IR spectra of SN 1994D reveal a flux `deficit' in the R, I and J bands as early as maximum light. They also illustrate the dramatic deepening of the J-band deficit after maximum light. We also present a maximum light IR spectrum of the peculiar Type Ia SN 1991T. This also shows a deficit in the J band, but it does not show such a pronounced deficit in the R and I bands as SN 1994D. Both supernovae show a P Cygni-like feature with the absorption at ~1.05 μm and the emission at ~1.08 mum. In SN 1994D the absorption shows no wavelength shift during the period between -8.5 and -1.5 d before maximum light. After this the feature rapidly weakened. We argue that, in this event, the P Cygni line feature formed in a discrete shell-like zone lying above the photosphere. In SN 1991T a continuous, shallow density gradient scattering zone seems more appropriate. For both supernovae we explore possible identifications with HeI 1.0830 μm and MgII 1.0926 mum, but there are difficulties with either option. In SN 1994D it may be that the feature is actually the result of a blend of the helium and magnesium lines. However, for both supernovae it is not ruled out that the J-band feature is really the result of a transition in one or more unidentified species. Consideration of the velocities associated with the feature indicates that, if its origin is helium, then in SN 1994D it is more likely that it formed in the alpha-rich freeze-out, but some mixing to higher velocities is also required. In contrast, in SN 1991T we favour an origin in the form of a residual or accreted layer on the surface of the progenitor white dwarf. If the feature is due to magnesium then the derived velocities for SN 1994D are in good agreement with the predictions of explosion model W7. In SN 1991T, identification with magnesium presents problems in accounting for the inferred velocity structure.