Upon opening up one of the units (a LCD monitor), several of the capacitors look fatter than they were supposed to be. The tops are conical instead of flat. I opted to just replace all of the capacitors, making sure I got one with the same (or higher) voltage and temperature rating. As soon as it was replaced, it turned on easily.
Another unit came from a former co-worker. It's a Thecus N5200 NAS box. This unit, like one of the LCD screens, keeps blinking when you plug it in. The owner was told that it must be the motherboard and that he should ship it back for replacement. He was also assured that his data could be retrieved if he hooks up his drives to another unit. Because I have the exact same box, we were supposed to hook up his drives to my NAS and recover his data, as he does not know how long it will take a replacement to be sent.
Before doing anything, though, I wanted to see for myself what the symptoms looked like. So I hooked it up to a powerboard and plugged it in. Like the LCD screen it was blinking on and off. I said, this is more likely a power supply problem rather than motherboard. I pulled out my power supply and swapped it in, and the unit came back to life. We left it plugged in and started copying data off the drive.
While it was doing that, I asked to have a go at the power supply. I would have to break some seals and since warranty is long expired, there is no harm in trying. Off came the screws. This was a very compact power supply and very different from standard ATX (Although it did have a standard ATX connector). I had to unscrew 2 huge heatsinks and revealed the capacitors. Sure enough 2 of them were "bloated". I took some capacitors from my spares bag. One was an exact replacement, the other was not (electronics store already closed). After taking a long time to put the heatsinks back, I plugged it in. No clicking sound. That was a good sign. we plugged it into my unit and it came to life, booted up. We then decided not to continue copying and just put the whole thing back together. It's been over a week and I haven't heard anything about it so I'm assuming it is still working.