Paintballers are born, no question. But then again, so is everybody else (except perhaps Robbo, who was designed by the secret test labs of the KGB).
Seriously, like a lot of other sports, most skills used in paintball have to be taught. But there is also the presence (or lack of) 'God given' talent. Some people, no matter how much training they put into it, will just not get good at something. Take running fast for instance. A lot of people do not realise that the ability to run fast is a combination of talent and skill. There are ways to utilise your body in order to run faster. Some people who are not that fast can become really fast after working on their technique, some people can not. This can be because they have a poor bodily coordination, something that you are born with or without. Through training these people may become proficient at running, but they will (almost) never excell at it.
Likewise, someone who has the talent to run fast, but does not use this talent, will never be as fast as someone who has mediocre talent, but does train in running.
So it's a combination of natural talent, which is probably better called potential, and training.