No, but technically yes. Only through cheating devices, however.
For example, I have a level 250 Eevee on Gold. But when I look at him on Stadium 2, it says he's level 31-ish. As soon as I level him up, he turns back to 31. Yeah, it kinda sucks.
So, I suppose you could have one to brag. But you could never use it, otherwise it'd go back to it's original level.