If Windows 7 can't, would Ubuntu be able to do it?
This is bios dependent rather than OS dependent, as once in hibernate or shutdown, the OS is no longer in play.
Check your Power settings in the bios for wake options. If it has something like "Wake from Keyboard", or "Wake from USB" or "Wake from Legacy USB", then you can give it a try. It is not always present because the USB controllers are often powered down during hibernate and poweroff so there is no way for them to signal. If you have a PS2 keyboard you may have better luck.
(this is all assuming you aren't talking about a laptop :))
short answer -- no
hibernate and shutdown both mean that the machine is off. the difference between hibernate and shutdown being that before the machine shuts down in hibernate mode, it writes it's state to the hard drive so that it can retrieve it on startup. so the only thing to turn on an off machine is push the power button.