The rapid rise in resistance occurring in barely conducting quench-condensed Pb films cooled through temperatures characteristic of the bulk superconducting transition is found to be strongly current dependent, the resistance increasing rapidly with decreasing current and temperature. Annealing the same film at temperatures below 40 K changes the behavior to that of a conventional superconductor with resistance that drops as the film current and temperature decrease. Experimental evidence suggests this results from a transition from quasiparticle-dominated to Josephson-dominated tunneling.