Heater will light up and then it just cuts off. Pilot stays lit.

Make sure thermocouple tip is seated inside the burner flame.
Make sure thermocouple is not overtightened. Finger-tight + 1/4 turn.
If the pilot lights, then ECO energy cut off is not tripped, but gas control valve might be bad.
Check color of pilot flame: it should be strong and clear blue.
If pilot flame is weak, or color of flame is yellow-orange: Clean pilot orifice and burner and combustion parts on yearly basis.
Clean air intake screen if applicable.
Increase air supply to water heater. Open a window and see what happens.
Check for adequate draft going up chimney using match under vent hood to see if smoke drafts immediately upwards.
Bleed air out of gas line.
Check gas pressure.

Open following links for troubleshoot resources:
http://waterheatertimer.org/pdf/Pilot-light-will-not-stay-lit.pdf
http://waterheatertimer.org/pdf/Pilot-outage-troubleshooting.pdf
http://waterheatertimer.org/pdf/Vent-spillage-test-atmospheric-gas-water-heater.pdf
http://waterheatertimer.org/How-to-install-gas-water-heater.html#troubleshoot

If I understand correctly, the pilot light is not staying lit.
You light the pilot and water in the tank reaches temperature.
Once hot water is gone, then pilot light has to be lit again.

Gas pressure is good, and gas is flowing.
Thermostat is working at high end because it cuts off heating.
Vent is working because tank heats up.
Air intake is working.

Gas control thermostat might not be reading low temperature. Replace gas control valve.
Pilot tube might be dirty. Clean burner parts and combustion chamber, and air intake, and vent.
Backdraft might be blowing out pilot. Check for constant updraft by running water heater 5 minutes, and then light match under vent hood and see if smoke goes up.
Very hot air can put out pilot, especially when water heater located in attic. Add ventilation.
Improper vent might have water dripping back down into combustion chamber and putting out pilot light and rusting out the parts. Inspect for rusted parts and water.
http://waterheatertimer.org/How-to-install-gas-water-heater.html#troubleshoot
http://waterheatertimer.org/pdf/Pilot-light-will-not-stay-lit.pdf

there could be a small blockage in the pilot light gas nozzle which just need cleaning

Try cleaning the orfice for the pilot light. You will have to dis-assemble the fuel lines from the thermostat to the burner to get at it. The orfice has an extremly small hole in it, do not make it any larger, just clean it out.

thermocouple probablly bad, is the flame covering the thermocouple.
the flame has to hit top portion of thermocouple to work correctlly.

It sounds like your main valve. The diaphram may be loose fitting or it has a small leak.

Cleaned pilot light area with an air compressor. Problem solved!

replace the therocouple. next year do not turn the pilot off

crappy pilot flame (not properly heating the thermocouple), bad thermocouple, thermocouple loose where it screws into the pilot safety valve, bad safety valve magnet.

One or more of those is the problem.

Oh...there is also a saftey switch...in case it tips over. It will also cause the pilot to not stay lit (if the switch or wires that go to i are faulty)