Entirely possible... in WWII. Which is what he would have grown up hearing and believing. Therein, lies the reason he thought he could get away with it.
Somewhere between Korea and Vietnam, that kind of promotion ceased to exist.
In the Marine Corps, the Mustang program, now exists to promote enlisted Marines to either Warrant Officer or Commissioned Officer ranks. In order to qualify, today, a Marine must be a Corporal, or higher, with high marks and test scores before being accepted. Graduate with a Bachelors and, I believe, go to OCS.
I was accepted to the program prior to being medically separated. Something that today, with a bit more wisdom under my belt, I regret not pursuing more vigilantly and ignoring the Navy doctors a smidge more. Had I fulfilled that obligation, that process alone would have taken me the four years of schooling and I would have owed another six years of service as payment for my education. My total time in service (TIS) would have been roughly fourteen years at that point and I would probably have been a Captain at best.
I just don't see a Vietnam era PFC getting enough attention from the higher ups to be able to be promoted in this way. I certainly don't see how in nine years even a 2nd Lt. could make full bird Colonel, much less a PFC.