PUSH THE ENVELOPE
From Claire Walsh (related questions came from Clive Pullinger, Linda Webb, Brian MacWhinney, Sue McCoy, and others): "I've always been puzzled by the phrase pushing the envelope; it's an incongruous image that doesn't seem to have any relationship to its meaning. Can you tell me where it comes from?"

It comes from mathematics, specifically as it is used in aeroplane design. It was popularised by Tom Wolfe's book of 1979, The Right Stuff, about test pilots and the early space programme. It's an excellent example of the way that a bit of specialised jargon known only to a few practitioners can move into the general language.
In mathematics, an envelope is the enclosing boundary of a set or family of curves that is touched by every curve in the system. This usage is known from the latter part of the nineteenth century. It's also used in electrical engineering for the curve that you get when you connect the successive peaks of a wave. This envelope curve encloses or envelops all the component curves.

In aeronautics, the envelope is the outer boundary of all the curves that describe the performance of the aircraft under various conditions of engine thrust, speed, altitude, atmospheric conditions, and the like. It is generally taken to be the known limits for the safe performance of the craft.

Test pilots have to test (or push) these limits to establish exactly what the plane is capable of doing, and where failure is likely to occur - to compare calculated performance limits with ones derived from experience. Test pilots called this pushing the edge of the envelope in the 1950s and 1960s, but this was soon shortened.

Following Tom Wolfe's book and film, the phrase began to move out into the wider world; the first recorded use in the more general sense of going (or attempting to go) beyond the limits of what is known to be possible came in the late 1980s.