Recently we have been getting power failures due to storms and what have you. Afterwards the computers will not turn on unless I unplug the power cable and plug it in again. I don't even have to let it sit; a simple unplug and back in works. The computers will turn on and be fine. I thought I heard the term "crowbar" describing some type of short that a computer can have which requires unplugging all the cables and letting it sit for awhile.
IS it plugged into a surge suppression unit / power supply protector? This can help and also it alerts you if the system is shorted out via hte power light and there is a simple reset button to push if it does short out on you.
'Crowbar' refers to an electronic circuit inside the power supply which is designed to protect the components the power supply feeds (i.e. the rest of your PC). When the circuit detects a surge or rovervoltage coming out of the power supply it immediately short-circuits the output (hence the scary word-picture in the name 'Crowbar' of someone literally throwing a physical crowbar across the terminals). The result of this is that the power supply immediately shuts down and will stay shut down until you remove and re-apply power, resetting the crowbar in the process.
With a dodgy mains supply, particularly where storns etc are involved, it is quite possible to get spikes and surges where the mains voltage goes up as well as down, so what you have described sounds feasible. There are a few things you could do about it - you could use surge-suppressing plugs or mains socket adapters on the affected machines, you could use a good UPS (which will smooth out the surges and dips) or you could look into a bigger UPS/power conditioning solution for the whole of your office