A The New Yorker article, by Nick Paumgarten
I first read this article, about someone who got trapped in an elevator (lift) for 41 hours, then I couldn't find it again. I just found it, by remembering the phrase "most over-elevatored building in the world". I was looking for this article again for this citation:
"... elevators (...) are typically borne aloft by six or eight hoist cables, each of which, according to the national elevator-safety code (...), is capable on its own of supporting the full load of the elevator plus twenty-five per cent more weight"So - at least 6 cables, each of which can hold more than the lift's maximum weight. In addition to these, lifts have brakes. There's really no reason to fear falling lifts. I think all the fatalities from lifts have come from getting caught in the doors and the lift moving, or similar events (barring building collapse or planes crashing into buildings). Some pets have been killed when they were trapped on the opposite side of the doors from their owners, and strangled by their leash.
Update 26 Jul 14
Another article, by HowStuffWorks, also says elevators have 4-8 cables, and each cable is capable of supporting the entire lift car.
"What if you were on an elevator and the cable broke?" 28 April 2005. HowStuffWorks.com.