We have numerically integrated approximately 500 systems of mutually gravitating bodies which were based on subsets of the uranian satellite system. In each run within a set, the satellite masses were initially multiplied by a common mass enhancement factor mf. The simulations were terminated at the "crossing time," tc, when mutual perturbations excited eccentricities sufficiently large for orbits of a pair of bodies to cross. For a given set, tcis well represented as a power law function of mfof the form tc= β mfα, where the values of the constants α and β depend on the system; values of α ranging from -13 to -3 are found here. This mass-scaling relationship may have wider implications as a diagnostic for the stability of many orbital configurations. We find that satellite systems which orbit around a significantly oblate planet are slightly more stable than identical systems in orbit about a spherically symmetric planet, presumably because the precession induced by planetary oblateness precludes secular resonances between the moons. Extrapolation of our results suggests that the five classical satellites of Uranus are stable over the age of the solar system (in the absence of tidal torques from the planet). Uranus' inner moons appear far less stable, with Desdemona conceivably colliding with either Cressida or Juliet sometime within the next 4-100 million years (provided the satellite masses adopted here are within a factor of 2 of the correct values). Thus, at least some of Uranus' inner moons are probably "young" by geological standards. Implications for the origin and evolution of these satellites are discussed.