Know that my post speaks not of the naturally gifted, but more to the idea that rules don't have to be treated as "rules" and there's an inherent danger in feeling the need to learn them first as rules, because it then begs this very question, "When is it OK to break them?"
Understanding the value of good composition is extremely important, but it's sort of like learning math - it's either going to cripple you because you really don't understand it completely and can't possibly figure out how you're going to apply it in real life, or it's going to be a fundamental undercurrent to whatever you do that you barely have to think about. The more you expose yourself to good composition the more you're going to understand the way in which "rules" were applied once you learn them. My admonition is to make the study of great works a priority over the studying of the rules, for as you learn the math behind what makes for a great photograph you will be able to apply it to the images already in your mind's eye. If you learn the rules first, I posit that most people will go searching for what fits the box in their mind instead of approaching subjects openly and finding the composition that lies beyond a particular rule. I tend to see far more photographs that are compositionally correct and utterly lack in impact than those who grab your eyes and won't let go, and yet require you to think hard about just what compositional rules could be applied to this amazing photograph.
There are many other rules that go into good composition than the Rule of Thirds, but because it's the one that's easiest to understand it's the one that is always taught first, and it's a boat anchor to so many photographers. I think it's far more important to learn about the use of light, color and contrast in composition, but they are the ones that take a lot more practice and study, and you can't simply choose an overlay in your editing software to see if you have it right. I just bang my head against the wall every time I see a wonderful photograph out on a forum and the first comment is chastising them for not having the horizon on a 1/3 line, because the commenter is reacting to the "rule violation" and not the art staring them in the face.