Classically, comets from the outer solar system (beyond the orbit of Neptune), are expected to be icy, and thus active near the Sun, while asteroids in the inner solar system (interior to the orbit of Jupiter) are expected to be relatively ice-deficient, and thus inert. Studies of anomalous objects, most recently 133P/Elst-Pizarro, challenge this classical picture, however, and suggest that either (1) subsurface ice can in fact be preserved over billions of years in small bodies in the inner solar system but still be close enough to the surface to be excavated by an impact by another body, or (2) non-gravitational dynamical evolution (primarily driven by asymmetrical outgassing) of icy bodies from the outer solar system can drive these cometary bodies onto thoroughly asteroid-like orbits, erasing all dynamical signs of their trans-Neptunian origins in the process. The question thus boils down to whether occasionally sublimating icy bodies on stable asteroid-like orbits in the inner solar system, particularly in the main asteroid belt, may in fact be native to the region or whether they must necessarily be recent arrivals.