Steven,
I've had my iBook a month now too and the same thing has happened to me twice, both times I was running off the battery (not the power adapater).
The first time it just cut off, and as you say, no shutting down, just dead. I thought maybe the battery had run out (even though I knew I had about 70-80% charge)but there were 3 green lights on the battery indicator.
After about 10 minutes trying to turn it on with nothing happening, it finally switched back on and all was fine again.
The second time happened yesterday, 5 days later. This time I was at home and had a full charge (and the battery indicator on the back showed 4 bars). I tried everything to turn it back on - connecting the power adapter made no difference nor did pressing Ctrl, Apple Key and power button. I kept trying it every couple of hours but no joy. As a last resort I took out the battery and put it back in again and it worked!! Unfortunately I had lost the slideshow I was working on and the clock was reset to 1970.
So my diagnosis is that maybe there is fault with the battery connections. The first time it happened I must have nudged the battery a bit which made it work again. Though this doesn't explain why the battery indicator on the back still worked?
I'm going to call Apple Support later - no doubt they will make me spend hours reinstalling everything and I'll have to wait till it happens a third time before they'll do anything.