Problem with most people, especially smart ones that think they know how the world works, is that they just don't know when ESP hits. Yes, there is precognition. Yes, ESP, psychic communication between minds--this stuff happens all the time.
The problem with ESP is that it really, really seems like you're just thinking. You're basically using your "organ of thought" or mind or something in multiple ways. And to most people, this will seem like coincidence when they notice it. And they will not notice it most of the time. If you pay attention, the coincidences start stacking up until they're no longer coincidences.
It took me a few decades to go from a disbeliever but avid sci-fi fan of ESP to actually discovering that I had ESP and basically everyone else does too. It's subtle, only it's not that subtle once you've been paying attention for a long time. Actually, it's still subtle then. I mean I doubt this kind of thing all the time, but presumably others who aren't as noisy in their own heads (reading slashdot and thinking and shit ;)) have it easier. I've certainly known many.
The short answer as to how this stuff works is that when you're thinking, using the parts of your brain that are built for concepts, math, tech, science, logic, etc., the part of your brain that can do ESP is no longer active. There's been some interesting brain imaging studies done, tons of them, that basically show this. It explains why doubters and especially scientists (!!) do not experience ESP.
Look, just as far as precognition goes, I have had all kinds of precognitive visions and dreams and shit. And this is not unusual. The only unusual thing about it is that I go ahead and assume that some of that is precognition. I have found, and a whole lot of other people have found this too, that if you regularly write down your dreams, you will find many which come true--many very specific and weird things, combined with other random stuff, that is so much of a coincidence, and it keeps happening, every other night, that precognition starts to look like the only reasonable explanation. I'm sure 95% of you will either try to mod me down or scoff! :) But look, I do understand chance and the odds and all of that jazz. I know when things are just coincidence, and when the coincidences happen over and over and you realize you're just calling them coincidences because you're using an outdated model of how everything works... then you realize it's synchronicity or precognition.
And then there's the psychic connection, talking between minds. That's pretty regular and easily missed. It's what's happening when you start thinking about a good friend or family member right before they call or email you. It's really noticeable when you start thinking about people you haven't talked to in a year and then go check your email and find an email from them. Like I said way up top, this info comes in looking like your own thoughts, with slight differences, and also seriously throwing doubt on the whole nature of thought itself. But it's like a seriously confused sense--some info coming in and you just haven't been trained to understand it yet. Like being a baby before you understand how vision and hands and all that crap works. It takes time for the brain to sort it out, and it also takes feedback. So our ESP senses are basically never talked about, never practiced, never shown to us by our parents, our schools. It's a skill, a language, an organ, take your pick, that we are essentially all retarded at, but not because we don't have the innate ability. Our society just hasn't figured out that it exists. And yet, quite a few in our society have. Just not the mainstream scientists and pop culture.
I'm rambling a bit and I'm sure this will be modded down when people think I'm full of shit and crazy... obviously.
I wish that you guys, whoever makes it this far, would maybe trust me when I say there's a dozen experiences in my life that absolutely have no other explanation than ESP (or crazier magick, creating stuff out of nothing skills, so ESP is the simpler option here). There's hundreds more that I think look like it easily, but I just don't have time to spell out the top stories that left me with no doubt. I remember being on the other side of this equation, being in awe and serious doubt about anyone claiming to be able to do this, but then I met dozens of people who became friends who experienced this stuff, I began to notice it and experience it. I knew people whose minds were so still (buddhist monks) that they would regularly communicate via ESP. I knew people who would be dead now if they hadn't paid attention to a precognition and slightly altered their behavior (like buckling up before a major accident). Hell, I would be dead now if it wasn't for two separate "coincidences." And yet I ignore ESP *all the time*. It's still subtle. And most people in our society think it's either bullshit or God/Jesus talking to them (which it ain't).
So yeah, it's real, real weak, like the parent poster said, but only if you don't believe in it and don't pay attention to it. You can learn to use it and develop it like any other sense or skill. Going into it by thinking a lot is a surefire way to not experience it. Yet, it will leak into your thoughts all the time. Just try out the hypothesis that not all of those thoughts are actually originating in your own head, or at least not by that thing you think of as your self. Which sounds crazy, but seriously, the key to actually experiencing ESP is to first believe it's possible. Then you realize it's always been happening. You just didn't notice it. It's called having an open mind. :)