There are stories about people using coils under high voltage transmission lines to "steal" power. It could probably be done but would get tricky depending on the load on the lines at any given time. I can say that I have taken an eight foot fluorescent bulb out, stuck one end into the ground under a 250KV transmission line, at night, and the bulb lit up, full brightness.
I have a similar problem here but it's RF related. There's a 25KW commercial, AM, broadcaster about two or three miles away, as the crow flies. I had to put bypass caps on all of the X10 switches to keep the pilot lamps off when the switch is off. The RF has even been enough to make LED floodlights pulse fire as their caps get charged from the RF field. Stray power, like that, can be a problem. Everything metal in substations is grounded to protect the people working there from induced voltages like that.