Totally believable. There's always been a universal trend during wartime to promote what works, not what might be officially sanctioned (at least among successful armies, which is one factor that helps make them successful). Some (but certainly not all) reasons for a hastened climb up the ranks include:

-Technical training: For better or worse, people with specialized training, education, or experience tend to be promoted faster. My grandfather was drafted about a year before Pearl Harbor, was a Staff Sergeant when the attack happened, and ended the War as a Master Sergeant. He turned down several opportunities to receive a commission along the way (as he put it, "I'd rather be at the top of the bottom than the bottom of the top."). All of this was due to him having gone to a technical school instead of a normal high school.

-Success in Combat: Also known as battlefield promotions, or battlefield commissions. During WWII a lucky or gifted Private could suddenly find himself promoted on the spot, even to an Officer rank (a feat that nowadays requires a bachelor's degree and additional training). As I said previously, promoting what works.

-Perceived Age/Maturity: Again, for better or worse, soldiers that are perceived as being more able to handle the stress of command will usually find themselves being given preference. The reasons for this are fairly obvious.

-"Needs of the Service": Battlefields are messy and chaotic places, entirely devoid of the logic and norms of the civilian world. The powers that be have to try to get results with what is available, and this can often lead to promotions being given for surprisingly arbitrary reasons. I've read accounts of soldiers being placed in positions of authority based on things like "being tallest", "being the first one in line", or even because they were the first one "the Sergeant noticed". You're an exhausted and stressed Platoon Sergeant that has had two-thirds of your unit wiped out in a few days. You've been ordered to recommence your assault on the objective, and have been given a gaggle of nervous and green replacements to complete the task. Do you really have time to conduct interviews and compare service records? Or do you just point at someone that looks like they might be able to survive the next few hours and inform them that they are now a squad leader? I remember reading about one Marine unit in the Pacific that actually had two sets of leaders. One was for the battlefield, and the other was for the drill-field. When they were preparing for an assault they promoted the first set, and when the battle was over they would remove their chevrons and hand them over to those who knew how to do drill.

As a final note, the reverse of this could also be true. There were some people that no matter how hard they tried, or how well they performed on the battlefield, never seemed to rise the ranks. I believe it was from the same source as the "two sets of leaders" I mentioned above, that there was a Marine that would like clockwork get promoted, use the wealth from his promotion to get drunk, and proceed to do something that would get him demoted.