There are a number of reasons why someone might fail the PMP exam.
First of all, it's a difficult test. I don't care who you are, how smart you are, or how much project management experience you have. At the end of the day it's a tough test.
I have had a number of students come to our prep camps who have failed the exam prior to the course. Most of those folks considered themselves to be very knowledgeable about project management, good test takers, and quite a number of them actually held other certifications and credentials. However, when they got into the PMP exam, they soon found out that this was not a test you can logic your way through. In fact, years of experience and a dose of logic can actually be counter-productive on the PMP exam. That's not to say that there aren't those people out there who can self-study and pass... they definitely can, with the right resources.
So why do people fail?