Current-voltage characteristics in the insulator bordering superconductivity in disordered thin films exhibit current jumps of several orders of magnitude due to the development of a thermally bistable electronic state at very low temperatures. In this high-resolution study we find that the jumps can be composed of many (up to 100) smaller jumps that appear to be random. This indicates that inhomogeneity develops near the transition to the insulator and that the current breakdown proceed via percolative paths spanning from one electrode to the other.