Like, in general (not in AnotherGuy's example above, where it's the orc rolling it)? I don't know, that's up to the GM. Maybe they're cool with one Persuasion check unraveling years of diplomacy or detente between tribes or whatever. That's not how I'd approach it, but it's their prerogative. There's a wide spectrum of success, which can be inexact, along the lines of what Bill Zebub said above.
"The DM has to believe it" also expands into the DM getting to call the shots on difficulty, so if they don't like it, they get to wave a magic wand and say this NPC is morally and ethically dedicated to their belief on the matter. (I argue that players largely have that same power, just different tools.) It's on the DM to use that power responsibly, of course.