Your fridge has a dual solenoid water valve - one side controls the water flow to the dispenser, the other to the ice maker. It's possible that one side burned out while the other is still good.
Here's a photo of what it looks like:
Here's a link to the replacement part:
http://www.searspartsdirect.com/partsdirect/part-number/Kenmore-Parts/Refrigerator-Parts/218658000/0046/253/Model-25358682890/0582/0165000?keySuffixId=NA&productTypeId=x&searchModelNumber=25358682890&subCompDesc=Icemaker&brandDesc=KENMORE&modelDesc=Side-by-Side&blt=11Before replacing the part, make sure the solenoid is in fact, bad. Get a voltmeter, and touch the probes to the electrical contacts on the solenoid, like so:
If the meter barely moves, the solenoid is bad.
If both solenoids test out OK, the other possibility is that the water supply line is kinked or clogged. Also, if your water dispenser has a filter (also on the back of the fridge), it might be clogged. My money's on the solenoid, though.
Good luck, and if this helps, please my answer!
Also, thanks to Zenzoidman for the pics.