There are at least eight places in the Bible where it very specifically says you cannot predict when the end will come. For the scholars among you, here are the relevant passages:
Matthew 24:36, 24:42, 24:44, 25:13
Mark 13:32, 13:33-37
Acts 1:7
1 Thessalonians 5:1-2
Seems pretty specific to me. It's right there in the book, and no set of arcane calculations is required; The Bible and Occam's Razor are in agreement.
The Bible's really trying to say "live your life as though your judgement might come at any moment." Good advice, really - maybe people'd be nice to each other for a change. (And look what we did to the guy that suggested we do that. Sheesh.)
On the other hand, the "Rapture" is not specifically mentioned, and nothing that even suggests it is mentioned in the Gospels at all. There's a mention in the Gospels that a place is being prepared for the believers, which is consistent with belief in a heaven. Then there's at least one passage in Revelation and another in 1 Thessalonians that suggests that the believers will be brought to heaven first. Even so, these passages leave a lot more open to interpretation - far more interpretation than the rather obvious prohibition on knowing when whatever's going to happen will happen.
It seems to me the desire to have the End Times occur during one's lifetime is a bit egotistical, really. "I will be your guide to most important event in the history of human history..."...puh-leaze.
In short, Harold Camping believes what he wants to believe. He's free to believe he's important. I'm free to think he's Mr. Camping's tent is missing a few pegs. But, hey, whatever.