As others have mentioned, it is quite possible on most PC hardware, although currently not very likely (as vast majority of malware does not bother).
What others have said is not possible is however wrong. Software actually CAN wake up a computer that has been regularly powered off either via "shutdown" or "poweroff" commands (GNU/Linux) or clicking on "start" button and then "Shutdown" (MS Windows), or via manual press of power button.
The feature is called RTC wakeup, and it allows software to schedule wakeup at specific time of day. It is controlled by Real time clock chip (chip which keeps track of time while your computer is powered off, and runs off its own CR2032 battery).
If you run GNU/Linux system, the control of that functionality is provided by rtcwake(8) system command.
As a related feature, many computers also have a feature called Wake on LAN, which allows other computers and routers to power on your computer over wired ethernet network (note that this functionality has to be enabled on your computer, and whether it defaults to on depends on your BIOS).