A bad CPU fan will cause this behavior with some motherboards. Others just let the CPU fry. Power supplies can, indeed, cause these symptoms as well, as can a flakey ATX mother board. If it's AT, not ATX, it's definitely the power supply.
I use Super Power cases and power supplies. The cases are sturdy (a bit on the heavy side) and open with a single thumb screw. The power supplies have been consistently reliable. Power Supply problems are a real pain and I like to avoid them.
I am very fussy about cabinets on issues of ease of maintenance. I will be maintaining and upgrading the machines I build for many years. If the customer didn't want me doing the maintenance, they'd have bought something cheaper elsewhere.
I recently got an order for a desktop machine and cringed. While I still maintain some desktops I haven't built one for quite a few years because they're so difficult to work with. Nontheless, I ordered a Super Power desktop case and have to say this, they make the best desktop case ever made. Even easier to work with than the mini-towers.