We have a player that quite legitimately rolls crits with his PC's long spear pretty much whenever he needs to. There isn't any question of cheating; he rolls his dice on a flat-bottomed platter sort of thing, and can just pass around the 20 when it happens.
It's rather scary yet pretty cool. He's had a lot of do-or-die fights that have depended upon scoring that one crit, and sure enough, he gets it.
Perhaps coincidentally, this didn't start happening until the god he was loyal to contacted him and designated him as the god's prophet (his god was popularly believed be quite dead, and the Prophet was charged with the task of spreading the truth surrounded his god's "demise"); two sessions later, he got his first do-or-die crit in melee with a dragon. Poor dragon went from 2/3 HP to dead in a single round.
And it's happened like that ever since. He HAS taken Improved Critical to help the process, and there's only been one enemy I can think of it where it didn't work out for him (in that case, the monster only had like 10 HP left; he just had hit with a single attack, and missed on everything); I can't even say one fight, since he'd done it to a different enemy in that fight like two rounds earlier.
No one else in the party has pulled off the "dramatically appropriate" crit except for maybe once in the campaign. Hell, the player doesn't get these sorts of rolls with his other characters.
They only go the Prophet, who appears to have been blessed by whatever weird little godling reigns over table-top gaming.
Granted, it also seems specific to longspears in the game I participate in. I can think of another PC that critted whenever it was dramatically appropriate with his longspear (usually after he'd just given a flashy description of what he was doing) -- but then the player switched to the mercurial greatsword, and he quite literally never rolled a 20 on an attack roll ever again.
Except against another PC, but that's a different story.