I was using my Galaxy Tab to read some comics. I'm sure when I powered it down, the battery still had some juice left. When Cols tried to turn it on today, it would not power up anymore. I tried different electric sockets and plugging it direct to a USB port to no avail.
Finally, after reading some Android/Galaxy Tab forums, other users suggested holding down the power button for at least 10 seconds. Some suggested touching the screen at the same time. I don't know if the latter is just coincidental, but I pressed and touched at the same time and it worked!