Is it possible that faulty wiring could be the cause of a very high electricity bill?

Absolutely!

Assume that there is a fairly low resistance ground fault at some distance from a breaker.

Assume that it is the feed for a high current device that only rarely, if ever, gets used [so that the natural load failure doesn't draw attention to the problem!] like an electric back-up heat bank used as emergency heat for a heat pump in San Diego [where the heater never gets called on!]

Assume, say, that it is a 3 Ohm resistance ground fault, so there is an 80 ampere load at 240 volts, let's say, on a 100 amp breaker, so that won't trip. This gives you a power consumption of 19.2 kilowatts [watts = current in amps x voltage] an hour, 24 hours a day. That's 13,824 kilowatthours [kwh] a month [30 days]. At 25 cents a kwh the WASTED electricity would cost $3456.00 in just one month!

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In a way, yes. You are paying for energy measured in kilowatts. If the wiring is old and undersized you are paying for the heat generated by the wires themselves. If this is the case you have a serious problem because heat destroys wire and a fire can result. If this is not the case, usage is the only thing that causes an increase in kilowatts used. Another thing to look for, because I have seen it, is your neighbor in an apt. building "tapping" into your service? If you have a separate main breaker for your apartment only lock it off and see if your neighbors complain to the landlord.