I believe there is some actual brain wiring that goes into whether or not you can, even with training, be an artist. Just the basic ability to recognize things that are pleasant, or aesthetically appealing - some set of preconditions in the way your mind is constructed that allow you to make artistic decisions. Also, realistically speaking, you need to have the willpower and stamina to make bad art for a long time, and know that if you keep applying yourself and learning more that you will eventually create better art. This isn't something that goes away, ever. You will always find flaws in your work (or at least you should always try to - if you are trying, you WILL find them!), and you will always be unhappy with what you produce in some way or other. Compromises, mistakes, bad design choices or ideas, etc.
If you are incapable of making the most basic aesthetic judgments, or lack the drive to continue doing something you love even though others do it better, then you lack the potential to be an artist. That's that!
However, I believe that if you have the most basic artistic predisposition, and some level of reasonable motivation, there is no reason that you can't start doing great art in your 20s, 30s, or way past that. My mom just started painting 5 or 6 years ago, and she has improved dramatically with regular practice and application, and has gone from simply copying flowers and still lifes to some intuitive grasp and understanding of some of the more important fundamentals, and she is (don't tell her i told you) well over 50!