Most of the old manuals will tell you this is a PSU issue. I was having similar problems with a IIci many years ago and found a junk IIci (longtime members may remember the "Ouch Free IIci" I bought on Craigslist, which was "painted" with highlighter...I later sold it to JRL on here and he cleaned it up nicely). The power supply in the "Ouch" machine worked, and the machine in general was in good working order, but I decided to swap the PSU with that of my IIci. It didn't fix the problems I was having. Frustrated, I spent $30 and eBayed another PSU. No dice. A few years later, I got the logic board recapped, and that IIci has worked fine since.
This also brings back memories of a IIcx I came across in 2005. It seemed to be dead. I left it on a table overnight and didn't realize I had kept it plugged in. The next day, I went to work on other computers at that same table. The IIcx was still there, still plugged in. Suddenly, without warning, the IIcx came to life, as though a ghost had pressed the power key. This was, no doubt, a capacitor related issue (and is something seen on Color Classics if their analog boards go bad).
The PSUs in the IIci/IIcx are quite sturdy. It's these boards that aren't. (Notice a trend--the higher powered stuff from the late 80s had the worst boards...while the SE and Plus can go on forever).