I had the same problem with my GE Profile water cooler - here's what I did:
Please read completely first before attempting!
Unplug from outlet.
When the bottle is empty, remove it. Twist the plastic collar that the bottle rests on about 1/2" and lift up and off. I found a block of ice completely choking off flow of cold water out of the reservoir and reducing flow to the others. This points to a faulty cold water thermostat.
Operate each dispenser handle to help empty the reservoirs.
Optional: This can be an excellent time to clean and descale the reservoir, tubes and valves. Bring the cooler outdoors, and remove the (2) screws securing the safety strap on the right-rear side - midway up. Unscrew the plastic cap, and the remaining water in the reservoirs immediately begin to drain completely. Replace the cap. Mix up a gallon vinegar & water (1 qt. vinegar to 3 qts. water). Pour directly into top to fill the stainless reservoir completely. Open each handle until water begins to flow. Let the vinegar an water descale the internal parts, valves and reservoirs for 30 - 60 minutes. Empty as described above and repeat. At this point, scale & mineral deposits should be minimal. You can either flush with 2 or more gallons of clean, warm water or disinfect by filling with a 10% bleach & water solution and it soak for 15 minutes, first. Operate the handles to flush and then remove the cap again to drain completely. Replace the cap & safety strap.
Remove the condenser coil (black grate) supporting screws and gently finesse the coil away from the unit enough to allow access to the top rear plate (be extra careful not to crush or otherwise damage the lines that supply this coil). Remove the screws securing the plate and remove it.
You will see (2) adjustable thermostats. The one on the left is for cooling and should have a the sensor or capillary tube running from it to under the Styrofoam insulation. The thermostat on the right is for heating and is not a concern (unless you are having issues with hot water). You may try adjusting the thermostat with a straight blade screw driver - twisting CCW lowers the temperature and CW increases temperature. If yours is like mine, it didn't help.
Remove the top plastic cover of the cooler.
You should now see the cold water reservoir wrapped in white Styrofoam insulation. Side and bottom seams are seal with wide, gray foam tape under a thin plastic tape. The thin tape will require a razor knife to cut and the then it and the gray foam tape can be pulled off.
Next, carefully clip the plastic ties that secure the rubber tube to the stainless tube at the top of the reservoir. Do not cut the rubber tube! You may have better luck using a pair of pliers and tightly gripping the tie mechanism and twisting to stretch & break away.
Gently remove the two half side sections of Styrofoam around the reservoir. Minimize disturbing the copper lines that extend out from the Styrofoam insulation. The sensor tube from the cold thermostat can be removed from the reservoir (save the white plastic sleeve and the curved copper sleeve over it) as they will be needed to be installed on the replacement thermostat. Both will slip off easily once hanging free. Remove the (2) thermostat support screws to allow the thermostat to hang free from the frame. The thermostat itself has just (2) connections - it does not matter which wire goes where - they can be pulled off the switch by hand.
I tested my thermostat by placing the end into the freezer and checking with a meter to determine when it would open (cold enough) and close (getting warmer). Typically, about 10 degrees between the two (35 and 45 degrees) is when it should open & close. Mine did not work this way and would stay closed too long (freezing the water) and open only briefly - to close again (never giving the ice time to thaw).
The thermostat used in the cooler is not directly replaceable by GE. GE had been sending the entire cooler to their vendor for warranty repair (hence no real replacement parts or schematics, etc. from GE). What is needed though is called a "beverage thermostat". Many HVAC/R supply houses sell only those with ridiculously long sensing tubes (8 feet!) compared to the 24 inch style installed in the water cooler. I found a great replacement with a 30 inch sensing tube for $18.98 (+ $5 shipping) You can also find them on eBay for $20 - $30 and up). Anyway, here's the info from my shopping cart at Zoro:
Zoro Item # G0446564
Product name: Water Cooler Replacement Parts and Repair Kits
Description: Cold Control Thermostat
Unit Price: $18.98
If the link above dies, try clicking: Elkay 31513C
The mounting holes are the same, the temperature adjustment is in the same place and has same orientation for increasing / decreasing the temperature. There are only (2) terminals for wiring (as mentioned above, there are only 2 wires and it doesn't matter which goes where) and the wires can be slipped on like the old thermostat; no cutting splicing, etc.
Carefully install the the sensor as the original. Be careful NOT to make sharp bends in the sensor tube. As the name implies, it is a hollow tube - sharp bends and crushing will cause it to fail instantly. The other copper tubes contain refrigerant gas. Treat them the same - bends, crushing and allowing gas to escape will cause it to fail.
Reassemble. Do not forget to install new ties to secure the rubber tube to the stainless reservoir tube. You can use duct tape if you do not have foam tape.
You may want to leave the bottle off and fill the reservoir manually. Doing so allows access to view the cold water reservoir so that you can adjust the thermostat so that ice does not form to the point of blocking water flow. Some ice is GOOD to keep water refreshingly cold, but should never restrict flow.
I hope this works as well for you as it did for me! Good luck!