It's got to be to do with one or more power cables short circuiting (wire(s) touching each other or the chassis) due to wires
touching together (probably disk drive power cables). This happened on my OLD Compaq Deskpro 386 that's now no longer around. I seen this happen on my OLD Compaq Deskpro 386 before. Make sure ALL the wires' insulation is not exposed!
It appears that it does this for safety feature when a cable is shorting: PC does not power on and the lights blink. That's the same symptoms the Compaq Deskpro 386 had. Or a exposed wire is touching the metal chassis. Make sure if that's happening to tape over that exposed part. Yes, I know you do need to open up that case at least one more time before more fun begins.

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