Quite the opposite. current would continue to flow in a burnt out cable.
As such what your are alos saying is that the current be high enough to cause an out of sight electric bill? To my knowledge the earth is a extremely poor conductor and would not add appreciably to the electric bill. Or would it just charge the earth around the faulted cable causing an electricution safety issue.
Going back to the OP, you still are of the opinion that a damaged underground cable is causing their high electric bills.