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{sub C} . Such massive NBWDs are hot (kT {approx} 100 eV), luminous (L {approx} 10{sup 38} erg s{sup -1}), 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{sub C} models. While SN Ia 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{sub C} .