Certain characters are introduced at the same rank they finished with (CPO Pertwee, Sub-Lieutenant Phillips), but others such as Commander, then Captain, then Commodore Povey and The Second Number One Lieutenant, then Lt Cmdr, then Commander Murray rise up the ranks as much as they would in the real navy. Notice however that the two straight examples of this trope are a petty crook and a cretin, respectively; who'd want to promote them?
Phillips was supposed to go before a promotion board in one episode, but ended up being court-martialled instead thanks to some kind of paperwork screw-up and nothing ever came of it.
And Pertwee was a career NCO. As a Chief Petty Officer, he is pretty much at the top of his career tree. At the time Chief Petty Officer was the highest NCO rank in the Royal Navy. The modern form of Warrant Officers (originally called Fleet Chief Petty Officer) were not introduced until 1973, and the older form was abolished in 1949. One could be commissioned from CPO at this time, but that was only available to certain branches.
This doesn't seem to fit how the trope is used:
Only Sane Man: Povey thinks he is; Commander Murray really is.
Among the senior brass, it's Admiral Ffontbittocks.
Moved these two items from the main page during a clean up effort. I'm not sure how to tackle either of them, they seem a bit discussion-y.
This doesn't seem to fit how the trope is used: