Class changes are established in canon. Tataru does it several times in the MSQ, as does Thancred. The latest expac trailer hammers it home even further with Derplander.
That being said, the question is how adept do you want your character to be known at different skillsets? Some players have every DoW/DoM/DoL leveled to 70 but still have their toon act IC like they have only one vocation.
My toon doesn't introduce himself as a DRK/SAM/MNK, but others know him as someone with a reasonable amount of skill at using either katanas or greatswords, and a family background of a grandfather who was a Fist of Rhalgr. So he's decent at punching and swords, with an IC flavor of using both in combat. He's no doctor or wizard, though - he asks others for their respective expertise in those fields.
Then speaking from the DM viewpoint, whenever a player asks me if they can try something from their field of expertise, I always allow the attempt - but I'm under no obligation to tell them the DC of what they're trying, or to hint that only a pure expert could do it. It will vary from clique to clique, but, for example, the healer who establishes himself as the healer who runs a clinic all day or is a combat medic for one of the Grand Companies, seems much more deserving of plausibility than someone who's 70 WHM on the side but never brings it up until that super special moment.