The term "Mad Minute" did not originate with the Vietnam War. It originated from British Army infantry skill-at-arms training from just before World War 1, when an infantryman - during what would today be called his Annual Personal Weapons Test, or APWT - had to get at least 15 hits on a 300 yard target in 60 seconds. No mean feat with a bolt-action rifle fed from a 10-round magazine normally loaded from 5-round stripper clips (and thus needing to be reloaded during the practice). This is why the German Army in the opening battles of WW1 frequently thought they were facing machinegun fire when they were mainly up against infantrymen with bolt-action .303 Lee Enfields on rapid fire.
The "mad minute" was used by, amongst others, author Michael Herr in "Despatches", in that case to describe periods of intense automatic weapons fire during the Vietnam War. Also used to describe any short period of frenetic activity.