Freezer on bottom works fine, fridge part dosnt

It sounds like the timer is going bad.... there is a timer that runs the fridge for 6 hrs. and defrosts for 20 mins. if the contacts are pitted ,it will run for shorted periods ..... this timer can be located anywhere from behind the unit where the compressor is to inside the regrigerator compartment ...

Just had the same problem and exact same symptoms with 21 cf Kenmore fridge w/ bottom freezer.  I ruled out problems with fans, etc.  Finally pulled off the freezer door, etc. to gain access to the evaporator assembly at the back of the freezer. You need to remove the sliding door/drawer hardware and ice-maker to remove the back panel -- takes a little time, but not a difficult job).  The evaporator was heavily iced up (maybe the defrost timer problem mentioned by Jerico, or maybe just an accumulation over a long time; a family member sometimes didn't slide the freezer door all the way closed).  In any case, the "snow line" had gotten pretty high, blocking off the refrigerator cold air return duct to the evaporator.  In the fridge compartment in my model, this is a small square grated opening in the bottom of the fridge, left hand side, back, under the left hand veggie crisper.  When that vent is blocked (can't see any problem from inside the fridge compartment), the cold air can't circulate back to the evaporator coils -- no air out means no air in, so no cooling in the fridge section.  Freezer OK because the evaporator coils are right there.  
Finished de-icing with a hair dryer, using small towels under the evaporator coils to catch most of the water.  I am about ready to fire it back up.  If the same problem returns in a few weeks or months, I will go for the defrost timer replacement.  If not, I'm done for a while.
Jim M