Dust inside the machine? Twice my A1200 refused to boot up properly (going scarily mad with the hard drive the 2nd time round- whirring in dagerous ways), but a small vacuum cleaner (like one of those portable ones) might do the trick. You'd have to open the amiga first (But-oh no!- that would void your warranty :-D ).
Another time it was the monitor. Check by connecting your Amiga to something else, if possible.
