  We present late-time Hubble and Spitzer Space Telescope imaging of SN 2008S
and NGC 300 2008OT-1, the prototypes of a common class of stellar transients
whose true nature is debated. Both objects are still fading and are now >15
times fainter than the progenitors in the mid-IR and are undetected in the
optical and near-IR. Data from the Large Binocular Telescope and Magellan show
that neither source has been variable in the optical since fading in 2010. We
present models of surviving sources obscured by dusty shells or winds and find
that extreme dust models are needed for surviving stars to be successfully
hidden by dust. Explaining these transients as supernovae explosions, such as
the electron capture supernovae believed to be associated with extreme AGB
stars, seems an equally viable solution. Though SN 2008S is not detected in
Chandra X-Ray Observatory data taken in 2012, the flux limits allow the fading
IR source to be powered solely by the shock interaction of ejecta with the
circumstellar medium if the shock velocity at the time of the observation was
>20% slower than estimated from emission line widths while the transient was
still optically bright. Continued SST monitoring and 10-20 micron observations
with JWST can resolve any remaining ambiguities.
