My Kodak z760 did the same. I thought about what it was doing and after it successfully powered up took a clean soft cloth to the lens barrel that extends on power up after it was fully extended. The problem vanished.
I have a Kodak CX6330 and sometimes when I turn it on, it looks dead and doesn't turn on. A red light flashes instead of the normal green indicator next to the viewfinder. When I put the button back into its place and then try switching it on again, sometimes it turns on but sometimes it doesn't. It can't be a battery problem because I tried a freshly charges NiMH battery. The NiMH is compatible with the cam and has worked before. Can you tell me what to do as it happens only sometimes and I will look like an idiot if I try to show it to a repair shop/center whatever.