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We present Hubble Space Telescope (HST) and ground-based optical and near-infrared observations of SN 2005hk and SN 2008A, typical members of the Type Iax class of supernovae (SNe). Here we focus on late-time observations, where these objects deviate most dramatically from all other SN types. Instead of the dominant nebular emission lines that are observed in other SNe at late phases, spectra of SNe 2005hk and 2008A show lines of Fe II, Ca II, and Fe I more than a year past maximum light, along with narrow [Fe II] and [Ca II] emission. We use spectral features to constrain the temperature and density of the ejecta, and find high densities at late times, with ne ≳ 109 cm–3. Such high densities should yield enhanced cooling of the ejecta, making these objects good candidates to observe the expected “infrared catastrophe,” a generic feature of SN Ia models. However, our HST photometry of SN 2008A does not match the predictions of an infrared catastrophe. Moreover, our HST observations rule out a “complete deflagration” that fully disrupts the white dwarf for these peculiar SNe, showing no evidence for unburned material at late times. Deflagration explosion models that leave behind a bound remnant can match some of the observed properties of SNe Iax, but no published model is consistent with all of our observations of SNe 2005hk and 2008A. [ABSTRACT FROM AUTHOR]