Why does my ceiling fan light turn off by itself?

I had a ceiling fan where the light turned off after it heated up, it turns out that the contact that the bulb screws into lost contact due to the expansion of the metal due to the heat. You could try bending the contact out a bit to see if that helps. Clean it as well while you're at it. Make sure the power to the light is off so you don't get a shock.

Another possibility is that the radio control is being activated by interference or someone else using a remote on the same frequency. Many devices like doorbells and garage door remotes use the same radio technology, so changing the radio channel your light uses may fix the problem. Your manual should show how to do that.

It's because you have the wrong wattage light bulbs in there. Take one out and it will come on and stay on.

This last guy is right. A lot of these ceiling fan lights have a wattage/voltage protectors. If the combined wattage of the bulbs exceed 190 watts, it will shut down the lights. Try using 40 watt bulbs, not 60. I just had this happen to me

I installed an outdoor Hunter ceiling fan 8 days ago (lantern model) that came with the 2 dimmable LED bulbs so the bulbs are not overheating. It also came with the 2032CR watch battery for the remote control. Occasionally when I use the wall switch to turn the fan & light on and off instead of using the remote control to control the fan and light on and off from it's previous setting I have that problem.

I opened the remote control, pressed the reset button and haven't had the problem the last 3 days.

I bought 2 Hunter Exeter model fans and was running into the problem of the lights cutting in and out after about 10-15 minutes of steady light output. Both fans were having the same problem. The fans are controlled by radio frequency remote controls - which the instructions say are "paired" with the fan when they arrive. BUT, you can re-pair them and doing so solved my intermittent light problem. I only re-paired one of the fans and now both fan lights work properly. My guess is there was some interference between the remotes for the two fans, and re-pairing one eliminated that interference.