The only thing I can think of is maybe the power adapter went bad on you? I know thats a tough thing to test too as its specific to the Cube (I have a G4 Cube too). Do you hear like the CD or DVD ROM make a slight noise (like the scanner slide slightly) when you press the power button? If you have speakers for it (USB only) do you hear the Mac chime at all? I know its a PITA to get another power adapter for it just to test and they're not going very cheap on eBay either.
Did you try a different outlet just to be safe?
Also, on mine, if you have it tilted slightly it will either turn on all by itself if its shutoff, or shutoff all by itself if its turned on. It must have a sensor somewhere inside it where if it gets tilted it triggers the power either on or off. Maybe its just mine, I don't know. So make sure its on a level surface with the slotload optical drive facing upward. Let us know how you're making out. Like I said I have a G4 Cube that I use at work all day long so if you have any questions please let me know.