In answer to the original question -- no. Subject, however, to one's personal judgement of what one can or cannot actually achieve. There is no point in trying to play something which is so far beyond either one's technical or musical skill that one can't make it sound decent at least to one's self. To play in public? Again, no -- but be aware that the public may expect, with some justice, a decent performance!
As to professional vs. amateur? Hooey. As someone has noted, the difference is in whether one gets paid or not, not whether one has a certain ability or not. Indeed I would go a bit farther than that, and limit the term professional to one who manages a sizable fraction of his or her income from performing or teaching -- but, except that that person is likely to be able to practice more (maybe not!) it makes no difference.
And as to conservatory trained vs. otherwise... again, I beg to differ. The major difference there is that it is likely that the conservatory trained individual has had access to better teaching. Whether or not they had the ability to take advantage of that is another question entirely, and it is perfectly likely that they may end up as a technically highly qualified musical snob with an emotional range all the way from a to b. Depends on the individual, not the name of the school they went to.