We investigate numerically the low-temperature equilibration of glassy systems via nonlocal Monte Carlo methods. We reexamine several systems that have been studied previously and investigate new systems to test the performance of such methods near the putative Kauzmann temperature, T-K, where the configurational entropy is presumed to vanish. Our results suggest that previous numerical claims in favor of and against a thermodynamic transition at a finite T-K must be reevaluated. Our work provides some guidelines and suggestions for future numerical investigations of disordered systems at high densities and low temperatures.