Recent high-resolution spectra of the Type Ia SN 2006X have revealed the presence of time-variable and blueshifted Na I D features, interpreted by Patat et al. as originating in circumstellar material within the progenitor system. The variation seen in SN 2006X induces relatively large changes in the total Na I D equivalent width (\Delta EW ~ 0.5 A in just over two weeks), that would be detectable at lower resolutions. We have used a large data set comprising 2400 low-resolution spectra of 450 Type Ia supernovae (SNe Ia) obtained by the CfA Supernova Program to search for variable Na I D features. Out of the 31 SNe Ia (including SN 2006X) in which we could have detected similar EW variations, only one other (SN 1999cl) shows variable Na I D features, with an even larger change over a similar ~10-day timescale (\Delta EW = 1.66 +/- 0.21 A). Interestingly, both SN 1999cl and SN 2006X are the two most highly-reddened objects in our sample, raising the possibility that the variability is connected to dusty environments.