No. Its ludicrous in any modern service area. The only place you've seen moves like this are 3rd world dicatorships where its handed out to the rulers immediate family like Kim Jong Un in North Korea. In Democracies its an impossibility becuase it just reeks of corruption.
As Captain of a Starship he's the modern day equivalent of an O-6. A brand new lieutenant its inconceivable even in times of great conflict for him to end up there after 3 years. Both logically and in terms of what you learn on the job. Take a great American hero of WWII in Dick Winters who was the prototype and lead for HBO's series "Band of Brothers". He starts the series as an O-2 (1st Lieutenant) and ends the series as an O-4 (Major) who likely would have been an O-5 shortly thereafter. Winters was a legitimate hero on order with other major hero's of the US and likely to be on a meteoric rise but he certainly wasn't just a cadet.
They "Typical" US length of time for moving from O-1 to O-6 is 22 years in the US and roughly 20 years in most other modern Military. Also that move at 22 years isn't a command necessarily though for some it is. That doesn't mean some people don't get promoted faster but 3 years is an absolute absurdity that really made me ignore most of the re-releases in alternate universe. The ONLY way this works is if you go back to non-standing Armies - Professional Services don't have these meteoric rises. The federation if they had no ships would have to pick a command structure that worked if say they had none and suddenly needed 300 , built them and had to staffed them yourself.
Non-Standing Army Promotion Examples:
Galusha Pennypacker is the youngest of the US Generals ever at the age of 20 , but he also started at 16. He recruited his own company to join the Union equipping and outfitting them initially (he came from wealth). However the reason for his promotion to General was that he was "mortally wounded" and his lifes goal had been to become a general in the civil war. (These was also the 4th time he was wounded in battle). Turns out it wasn't so "Mortal" and he survived it. Immediately after the war he was "demoted" to Colonel. He was also a Medal of Honor recipient and survived 5 gunshot wounds in several different engagements so giving him his lifes wish as he lay dying seemed like a good idea.
Personal: I saw this with the Iraqi Army as well after it was devolved and reconstituted. It was a total cluster, Colonels who couldn't lead their way out of a paper bag, Sergeants who were more poorly trained than American's just out of boot camp. Not a real "standing" modern Army which takes time to develop. I mean literally you could be under fire and the Captain wants to take a lunch break. It's a cluster that only happens in newly created forces (Note: I would have considered Younger Captains plausible in NX-01, since its a brand new fleet).
Why the TOS is plausible if meteoric:
Kirk's back story in TOS was a great back plot moment to me because it was at least plausible. He assumes command of the Farragut (a destroyer) when its Captain and First Officer are killed completing he mission. He's a Lt. Commander at this point (0-4). Presumably a year or two later he gets promoted by a board making him a full Commander. Promotion Boards in modern services love promoting someone "already doing the job well" So sometime a couple years later he receives a promotion to Captain despite his age.
From my own experience the "average" (typical) time in service from promotion to E7 in the Army is 13.5 years. Keep in mind some stay 20 and never reach it or reach it on their way out the door. I did it in 7 years 4 months but there were unique circumstances surrounding it. The fastest modern one is 6 years 3 months and its from the US special forces. 7 Years isn't unheard of certainly, but your talking about the fastest 4-5% of all promotions at that point. It comes from both hard working and the "opportunity" of being in the right place at the right time. In my case 30 days after I was promoted to E-6 I was in Iraq for the invasion. The Brigade I was in was convoying North, but was short E8's to lead certain sections of the Convoy. We had 4 E-7's in the section I took charge of but the Brigade CSM didn't trust them (2 were under investigation for misconduct , the other 2 were total clusters); I gave a really good briefing in my role of collecting intel on the planned route and he approached me afterwards about taking the lead NCO (command equivalent) with a Captain for this particular. So my very first look at the E-7 board exactly two years later there is a notation that as an E-6 performed an E8 role safely leading a convoy into Baghdad without casualty. Combine that with being the Honor Graduate at BNCOC which is the school that all E-6's have to attend and its an easy decision for the promotion board. Other E-6's may have years of competent experience but they didn't get the opportunity to fill in like I did. Didn't mean I didn't earn it but I wouldn't have been able to had the unit not been critically short.
Kind of Conclusion: I could see 30-32 if they had a backstory that involved two situations of remarkable opportunity. For example a Junior Grade Lieutenant taking command of a situation similar to the Siege of AR-558 in the Dominion War, hastily being promoted to Lieutenant , getting early look and making Lt. Commander. Then again, like Kirk getting command of a Starship through the death of its Captain and 1st Officer (and presumably the other Lt. Commander who outranks him). Consider how long it takes Worf to go from Lieutenant to Lt. Commander and how great he was as an Officer. The "Cadet to Captain" though is just total hokum unless your talking about a 3rd world dictatorship, volunteer fleet service (Starfleet's Civilian Readiness Corps maybe) , or a devastated people who just "have to put someone in charge".
