My tower has the on/off button, the reset button and at the back is also a switch - you might check for that.
or a switch might be stuck half way?

after a software crash or conflict mine will blue screen and pop up a message- but it keeps running until I do the restart

sometimes mine goes to the BIOS {internal system/motherboard program/settings} and then will do a reset/restart

but since yours totally shut off - it makes me think of a possible hardware problem - hardrive, memory, CPU, fan/power box??
- but usually some sort of message will come up -
unless the Power box has a problem- any odd noises- power box malfunction no power to run computer?

Do you have win XP? or what operating system do you have?