Misleading Japanese phrase: とてもじゃない (originally posted on FortuneCity)

Here's an uncommon but confusing idiom I've been misinterpreting until just recently: とてもじゃない, usually found in sentences similar to this one: 「とてもじゃないが、出来ない。」. What's so confusing about it? Well, here's what it looks like it ought to mean:
  • 「出来ない」 with no other context roughly means, "I can't do it".
  • 「とても出来ない」 is then roughly, "I absolutely can't do it" (literally, "very can't")
  • 「じゃない」 normally means "isn't" and the 「が」 particle indicates contrast.
  • Therefore, 「とてもじゃないが、出来ない」 would appear to mean roughly, "I can't do it, but it's not absolutely impossible" (can't, but not so "can't" that it's "very can't").

Unfortunately, that's not what it means at all. If 「とても出来ない」 means "I absolutely can't do it", then 「とてもじゃないが、出来ない」 means "I absolutely, positively, CANNOT do it". It's not a negation but an emphasis. The best explanation I've seen of where the confusing grammar comes from suggests that this usage is reserved for situations extreme enough that とても isn't nearly strong enough to use, so とてもじゃない isn't saying, "not very much so", but rather, " 'very' doesn't even begin to cover it".