My home computer is 8 years old and has been such a star, but is now refusing to start after being switched off every night.
EDIT: There's one thing I missed and didn't quite understand in your OP. "I got a rebuild to replace it". So, is it one computer we're talking about, or two different machines"?
If it's two different machines doing the exact same thing, then as Mr. Bazz suggested, it may be either a bad wall socket or power strip. Or alternatively, the same garbage OEM PSUs may both be faulty. The term "rebuilt" is somewhat nuanced, and means different things to different rebuilders. In many cases if a component was working while in their possession, then it stays in, regardless of its age, quality, or whether or not it's teetering on the brink of failure.
Then:
OK first off, if you really are "shutting off" the PC at night, IMO that's foolish for a number of salient reasons. If you're putting it to "sleep", forget I said that. Besides, the +5 volt rail is always on, even with the machine powered down. All you'd be saving is a few milliwatts of power, accompanied by the aggravation of having to reopen any folders, and restart your browser(s) and applications.
Enough of that. To rule out the case switch being bad, there are 2 pins on the wiring from the case switch to the motherboard, marked "power + & -". Pull those off and short out those two pins together with a small screwdriver, (and yes, the tip of a nail file will work in a pinch) Just absolutely DON"T touch any other pin or component in the process. Keep in mind the action of simply removing and replacing those wires might be enough to remove any corrosion present on the pins. In that case, the case switch could function normally when the wires are plugged back in.
There's almost no good reason to ever "pull the plug" on a computer. Under, or possibly next to, the power cord is a switch that turns off the PSU. By clicking it to off, there is no power going to the machine and the case is still grounded. So, if you touch any bare metal on the case, any static charge you might have will be sent to ground, after which you can safely work inside the case without fear of "frying any components".
Next, sometimes a system has to be "bled", to remove any residual charge in the memory. So, instead of pulling the plug, switch off the PSU, and hold the case switch down for a few seconds. Then, (obviously), click the power supply back on and try again.
You've received a considerable number of worthwhile and possible suggestions as to what could be wrong.
However, (IMO), we've missed one big one. The power supply, ("PSU") itself could be bad. Keep in mind an 8 year old PSU is not exactly a "spring chicken". Depending on which "rail" (or rails) is acting up, any number of quirky symptoms could manifest themselves.
There is also circuitry in the PSU that interacts with the BIOS and Windows "power management". If that circuit is faulty, it could create the symptoms you describe.
I had a "low mileage, one owner" Seasonic PSU in one of my machines that I put to sleep one night, and it just never woke up.
Not bragging, (well, maybe a little), I guessed PSU on the first try, replaced it, and lo and behold, we experienced a "let there be light" moment.
If any of this stresses you out, or you're uncertain of your abilities, remember, there's no shame in taking it to the shop. Cheers.