I guess first it depends on your definition of "professional".
For example, it has been stated many places that any person of average intellect/talent can, through intense study, reach master level. e.g. FM/NM. To some, FM level is "professional" level and you could, theoretically, make a living (although probably not a great one) at this level.
If you mean GM level, not so sure. Lazlo Polgar claims to have "created" GM level players from average kids (Susan/Judit/Sofia(IM, not GM)) through his experiment of home schooling and intense chess study from an early age. Whether or not it was his training regimen from an early age or talent in the Polgar sisters that created GM level players is unclear, though.
Another example I've heard of is Gata Kamsky. I've read that his father subjected him to isolation and intense study sessions from an early age. But again, it is not clear how much hard work alone contributed to his results and how much "talent" contributed.
Unfortunately, talent is a very subjective concept and hard to quantify very accurately. There haven't been many experiments in this area -- to get a handle on the nature vs. nurture debate in chess, you would need to test a large pool of kids with no prior study of chess for various factors that might show strong predictors of "talent" -- whatever those may be, e.g. spatial thinking, memory, etc. -- and then subject both a group of "talented" kids and "non-talented" kids (those without the indicators) to the same intense training regimen over many years.
I don't think there would be many volunteers for such tests, however. :-)
Realistically, is it ever possible to train oneself from an average skill level to achieve a professional standard of play?
I'm confident people can theorise and practice and raise their ratings by several hundred or so, but it seems some people have a natural flare for chess which cannot be replicated by study alone.
Is it ever possible to rival the phenomenal skill of inherent talent with simple self discipline and determined study? Does anyone know of examples?