It doesn't happen often, at least not since tires improved twenty-five years back, but it will happen at the worst possible time. The last time it happened to me I was 50 miles into a ride, which put me 35 miles from the nearest cell phone coverage. While inflating the second tube, I managed to break the valve such that I had only about 50 psi, and very slowly falling. Of course I hadn't checked the state of the vulcanizing fluid in my patch kit (bone dry), so off I carefully rolled towards home. My planned self-supported double became a century that included a bit of walking after the pressure got too low.