I just got these two systems that I was told blinked.  I cleaned one of them up all nice and put a game in.  But when I do this the light and screen flash for a split second, and then static, as if the system isn't even on, which appears to be the case.  The light stays off and there isn't any picture.  But if I take the game out it will work just fine, it'll do the blinking colors and solid color screen thing.  I tried the other system and it does the exact same thing.  I've never seen this problem before, and I've cleaned A LOT of systems over the years.  The weird thing is, before I cleaned the system I tried a game in it just to make sure it actually had a shot of working, and it "worked" fine, with blinking or glitched screens.  I'm not using a standard NES power supply, so maybe that's the problem?  Anyone else ever experienced this?

Have you used the power supply with another nes?  You could be right about the power supply.  There could be enough power to power it without the cart drawing power, but not enough power when the cart's drawing the juice too.  

Try another power supply or try that power supply in a nes that you know works...