  In a canonical model, the progenitors of Type Ia supernovae (SNe Ia) are
accreting, nuclear-burning white dwarfs (NBWDs), which explode when the white
dwarf reaches the Chandrasekhar mass, M_C. Such massive NBWDs are hot (kT ~100
eV), luminous (L ~ 10^{38} erg/s), and are potentially observable as luminous
supersoft X-ray sources (SSSs). During the past several years, surveys for soft
X-ray sources in external galaxies have been conducted. This paper shows that
the results falsify the hypothesis that a large fraction of progenitors are
NBWDs which are presently observable as SSSs. The data also place limits on
sub-M_C models. While Type Ia supernova progenitors may pass through one or
more phases of SSS activity, these phases are far shorter than the time needed
to accrete most of the matter that brings them close to M_C.
