there's a possibility it is low on refrigerant and the low pressure switch is cutting it off. Check for the A/C operation at the A/C amplifier:
http://www.camrymanuals.com/manuals/...nditioning.pdf pg 88
The A/C system pressure switch and cabin thermisistor come into this controller as well as engine temp ECT sensor and the output is the control to the A/C clutch relay. You can pretty well check A/C operation from there and then determine where the fault lies.
Electrical diagrams:
http://www.camrymanuals.com/manuals/99/99EWD.pdf pg 289
Don't know what engine you have, so there will be differences here.
The A/C clutch also has a 'lock sensor' that the A/C amplifier sees to make sure the clutch is engaged I guess. You need to check that too. It may be that the clutch is slipping and needs adjustment. If the controller senses this, it will shut the A/C down I think. As well, after a minute of A/C being on, feel the compressor and see if it has gotten really hot.
dave mc