bms1079,
Any of three simple possibilities come to mind - assuming the voltage to the power suppy is correct
(1) if the new case comes with a power switch (not the button, but the switch behind it), then it may be bad or miswired - you stated that neither the old nor new PS worked in the new case - you can check this by putting the MB back in the old case and using the old switch/power supply - if it works in the old case, then it is probably the new switch - if it doesn't work as before, you may not be connecting the power correctly to the MB or switch.
A technician would simply check the switch circuit with a continuity/ohmeter (no power connected) and check to see output voltages from the PS are correct on the output pins (power on)
(2) you may not be connecting the MB or switch power connector(s) correctly. Google the MB Model Number and look at the service or installation manual to verify correct power connections. Check the wiring diagram that came with the new case too.
(3) the PS may be bad, however you stated that neither the old or new PS worked. It is not likely that both are bad.
As usual, use common electrical and ESD (electro-static discharge) precautions when working around any sensitive electronics.
If this doesn't help, give us some details about the MB, case and PS (model #s, manufacturers.
Note: when you plug in power to the power suppy it is hot momentarily causing the fans and lights on some systems to briefly turn or flash. This is normal for most. The PS/MB then goes into a ready/standby state waiting for a signal from the switch to turn on.
Let us know what you find!
Dell
[It is usually the simplist solution that works]