Just had a desire to dig this up, throw a few of my own cents in.
They've determined part of the process that causes De Ja Vu (I believe it was a chemical, but I'm too tired to find the article and check). In tests, they created a false sense of recognition, and "bizarre feeling" associated with DeJaVu for completely new experiences.
Again, don't quote me, but I read somewhere else that the dream we have lasts only a few seconds out of our sleep pattern. So waking up at the end of it is either A) completely false or B) Not a proper dream, but a partly awake thinking state in which you're aware of your surroundings, and aware of the fact you're waking up.
Who says there's a difference between an amazing brain and premonition? Your mind is never aware of your thought patterns, so the ability to "guess" something a large portion of the time could be interpreted by someone who thinks more on the supernatural side as a "gift of clairvoyance" where as someone who takes the scientific approach would just consider it an accurate estimation.
If one knows all the factors of an environment, it is possible to predict exactly the outcome of any event. Of course, the accuracy of factors known is directly related to the accuracy of the prediction (for the simple minded - more factors, more accurate, less factors - less accurate). IMO, Chaos Theory only exists because we have difficulty in knowing all possible factors - if we did, I believe it would be null and void.
As an example, scrunch up a piece of paper, and throw it towards your bin at a high angle. You're effectively throwing blind, it's not just a straight throw but more chance - or so it seems. If you've got any ability, you'll make that throw over 70% of the time. Your mind is, without your input, calculating the angle required, the force required, wind resistance etc. Seems like guess work, but it's actually quite heavily considered. In much the same way, I can become Nostrodamus - seeing that in history as far as the eye can see, populations have increased, and with more interaction becomes more conflict - using this to predict world wars.
In my opinion, what is considered "predictions" are largely falsified in the brain - either we already know the info and allow/forget we knew it, so becomes a "prediction" that we already knew would happen, or false impulses created like Dejavu - or a highly accurate guess.
I apologise for this largely rambling sample of my thoughts, but I'm sitting at work with nothing to do (system issue), and had a few, party thought through brainwaves. Hope it actually comes across as a legitimate debate.
"The possession of anything begins in the mind" - Bruce Lee