Is it possible for a writer to overuse commas?

I have been afraid to use them ever since I took a college writing course as a freshman years ago. My professor and I were going over one of my papers in his office and he said to me a few times, "Why did you use a comma here?" Would a paper come off as more clear, concise, and compelling if you use proper grammar, but keep punctuation (such as commas) to a minimum? Is it possible to write a complex sentence without using commas and semicolons?