kenmore cycles

I am having an unusual problem with my Kenmore dishwasher. The unit is about 16 years old. It starts as normal and runs through the first fill, wash and drain cycle, and then when the timer switches to the next fill cycle, the motor hums, but the water won't fill again. If I turn it off and wait for about 1/2 an hour, it will run the next full cycle and then it won't fill for the next cycle again. I have checked the float and water valve/solenoid assemblies and they appear to be operating properly. The wiring appears fine and there do not appear to be any shorts.
Any suggestions would be greatly appreciated.

This, to me sounds like a water valve going sour. Sometimes they will work for a cycle or two and then just stop working. What I would do is, when you are not getting the water fill when you are supposed to, I would remove the bottom panel of the dishwasher and measure the voltage across the valve. If you are getting 120 volts across the valve and it is not working you know your valve is bad. If your not getting the 120 volts across the valve when its supposed to be filling, more troubleshooting is needed. If you are afraid of measuring live voltage you can test the valve for continuity by removing the wires and shutting off the electricity at the moment it won't fill. No continuity means a bad valve.

I checked the voltage this morning with the dishwasher running. When it was running and the water was filling, I was getting 120 volts at the valve. When the water failed to fill during the next cycle, I still had a reading of 120 volts.
As you diagnosed, it does indeed appear to be the valve that is malfunctioning. As soon as I can round one up, I'll replace it and hopefully the problem will be solved.
I appreciate your quick reply and advice.
Thank you.

Based on your troubleshooting, this certainly sounds like a bad fill valve. Good luck with the repair!