We have searched for ultraluminous X-ray sources (ULXs) in a sample of 28 elliptical and S0 galaxies observed with Chandra. We find that the number of X-ray sources detected at a flux level that would correspond to a 0.3-10 keV X-ray luminosity of similar to2 x 10(39) ergs s(-1) or greater (for which we have used the designation very ultraluminous X-ray sources [VULXs]) at the distance of each galaxy is equal to the number of expected foreground/background objects. In addition, the VULXs are uniformly distributed over the Chandra field of view rather than distributed like the optical light of the galaxies, strengthening the argument that the high-flux sources are unassociated with the galaxies. We have also taken the VULX candidate list of Colbert and Ptak and determined the spatial distribution of VULXs in early-type galaxies and late-type galaxies separately. While the spiral galaxy VULXs are clearly concentrated toward the centers of the galaxies, the early-type galaxy VULXs are distributed randomly over the ROSAT HRI field of view, again indicating that they are not associated with the galaxies themselves. We conclude that with the exception of two rare high-luminosity objects within globular clusters of the elliptical galaxy NGC 1399, VULXs are generally not found within old stellar systems. However, we do find a significant population of sources with luminosities of (1 - 2) x 10(39) ergs s(-1) that reside within the sample galaxies that can be explained by accretion onto 10 - 20 M, black holes. Given our results, we propose that ULXs be defined as X-ray sources with L-X(0.3 - 10 keV) > 2 x 10(39) ergs s(-1).