In the series, it's established that it's like Bella's mind is on a different frequency than what Edward is accustomed to. Ms. Meyer has stated that Bella can be influenced by other non-mind related powers such as Alice, who sees the future which is a "non mind" related power in the Twilight world, or Jasper's power to influence which is also "non mind" because it influences the body verses the mind - he uses the pulse and endorphins to influence.
Edward can't see into Bella's mind because her mind is so private and she is such a private person that no one can. I would think it would stand to reason that there are other people out there who are also similarly super private that Edward couldn't read but as most of his life is spent in two very small towns in Alaska and in Washington, he probably just hasn't run across any others.
Edward thinks his brain may be tuned to FM while Bella could be tuned on AM. In truth Bella is actually shielded and therefore he cannot read her thoughts which are further explained in Breaking Dawn.
Edward has always said that maybe her brain works differently than his and that he's vampire while she's human. Edward theories that his brain is probably tuned in on FM while her brain is tuned in on AM. I personally think that Edward is never able to Bella's mind because her thoughts are just simply shielded which discovered in Breaking Dawn that it's connected to her talent.
When Bella turns into a Vampire in Breaking Dawn they realize why, she is not able to be compelled by anything or anyone. She has the stongest powers any of the other vampires have ever seen. She is what they call a "sheild" she is able to resist the tempations unlike other "new" vampires which amazes everyone.
In the book Breaking Dawn, you realize that like Edward, Alice, and Jasper.. Bella also has a "super power." She is basically a "sheild" which she learns that she can push it out and protect others or push it out and let Edward hear her thoughts :)