I'll add my 2 penneth worth to this as I also have the same problem but on an Amilo A1645G. The lights are on but there's nobody home!
It seems to be intermittent still on mine. though it's getting worse. Initially it seemed that if the screen was opened up fully or (and this one's weird) the laptop turned upside down it would often spring into life.
What I have noticed is it that it doesn't seem to just be a case of the screen not powering up and displaying, as when the screen is blank it simply doesn't appear to boot at all.
A service manual would be nice, but they seem to be rarer than rocking horse ****
If anybody gets anywhere with this, I'd love to hear.