The faintest ultraluminous X-ray sources (ULXs), those with 0.3-10 keV luminosities 1 < L_X/10^39 < 3 erg s^-1, tend to have X-ray spectra that are disk-like but broader than expected for thin accretion disks. These `broadened disk spectra are thought to indicate near- or mildly super-Eddington accretion onto stellar remnant black holes. Here we report that a sample of bright thermal-dominant black hole binaries, which have Eddington ratios constrained to moderate values, also show broadened disk spectra in the 0.3-10 keV band at an order of magnitude lower luminosities. This broadening would be missed in studies that only look above ~2 keV. While this may suggest that broadened disk ULXs could be powered by accretion onto massive stellar remnant black holes with close to maximal spin, we argue in favor of a scenario where they are at close to the Eddington luminosity, such that radiation pressure would be expected to result in geometrically slim, advective accretion disks. However, this implies that an additional physical mechanism is required to produce the observed broad spectra at low Eddington ratios.
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