Is it bad to apply to the same PhD program two years in a row? I will be entering my first year of college next Fall. Even if most of my AP classes don't count for credit, I will probably be eligible to graduate after six 15–16 credit semesters. Though my desires might change throughout the next few years, I intend to pursue a PhD in Computer Science, Neuroscience, or something else that I'm interested in.

So, if I am eligible to graduate after three years, and I apply to and am denied from desired PhD programs, can I complete a fourth year of my undergrad and reapply to the same places?

I understand that each year of undergrad adds an additional valuable summer of potential REUs, internships, and other experiences. What are other reasons (besides the aforementioned and taking more credits per semester) people advise against completing undergraduate studies in three years?
You are absolutely allowed to reapply to schools in successive years. However, unless you have had a substantial updated in your profile, such as through improved grades, or additional work or research experience, I would not necessarily expect a better result.

However, it is definitely possible, and if there has been a substantial improvement, the results can be very different. (I've seen firsthand examples of this in students for whom I've written letters of recommendation!)