To take a dangerous risk, especially in conversation.
"You are REALLY pushing the envelope, trying to bone my sister!"
"You are REALLY pushing the envelope, and as a result, a plane is going to explode."
You'd be forgiven for thinking that the "envelope" in question is a paper one, but you'd be wrong. It's mathematical.
Now, you may think that by adding math to an article that's already about words and history that we've officially thrown ourselves into a perfect storm of perpetual nerd virginity, but first hear us out. Mathematics is used to calculate a plane's limits in a construct called its "flight envelope." That is, the particular combination of speed, height, stress and other aeronautical factors that form the bounds of safe operation. Go beyond these limits, or "push" the "envelope," and you'll be dancing into the danger zone where the dancer becomes the dance, and the plane becomes a giant fireball.
Test pilots, of course, have to push the flight envelope all the time. By flying a plane beyond its mathematical limits, they can check that it'll be safe when flown within its limits and find out what parts might fail if the limits are pushed, all while running the risk of having the plane burst into flames. There's also the rarely spoken about unofficial risk of having the moon collide with the plane as a result of the strong, gravitational pull brought on by the pilot's massive, massive balls.