To heat the overhead wires with electricity, you have to short-circuit them, which is both a waste and dangerous. (I know of someone who did it in a cold spell, though...) But serious icing on overhead wires is a rare occurrence and there are some devices against it. However, when overhead wires snap, it usually has to do with some defect. Most often it's some failure of the system to compensate thermal contraction (like a stuck roller), sometimes material failure, and sometimes something falling on or getting stuck in the wires.

Trains have batteries. They don't last forever, though, and in modern times, even a simple passenger car comes with a sophisticated software... which may be ill designed and shut down the system in the 'wrong' kind of emergency. (I could tell stories.)

"Catenary" is American English for "overhead wire"; though recently it is increasingly in use in rail literature this side of the pond, too (I prefer to use it, too).

_{*Lunatic*, n.
One whose delusions are out of fashion.}