We present new broadband optical photometry of Centaurs and Kuiper Belt objects from the Keck 10 m, the University of Hawaii 2.2 m, and the Cerro Tololo InterAmerican Observatory (CTIO) 1.5 m telescopes. These objects are believed to represent large protocomets in different stages of dynamical evolution. We find a wide dispersion in the optical colors of the objects, indicating nonuniform surface properties. The color dispersion must be understood in the context of the expected steady reddening due to bombardment by the ubiquitous flux of cosmic rays. We describe a model in which the dispersion is produced by stochastic collisional resurfacing. While non-unique, this model provides a plausible match to the observed color spread.