It's both. While there has to be an organized, test-one-thing-at-a-time approach for debugging anything more complex than a one-timer, art and experience come in when you have to figure out exactly where and what the error is. The more mistakes you make -- and fix -- the more easily you can recognize new ones.

Some people are so good at this, it appears to be magic. It's like the Knuth series -- though programming is essentially breaking a complex task down into little chunks and then describing them as succinctly AND in as much detail as necessary, it's a creative process. That's why it's an Art.