Could be excess humidity not allowing heat transfer and evaporation for the frost-free operation.
One thing that might help is get a refrigerator coil brush and pull off the bottom grill and brush/vacuum the buildup. UNPLUG the refrigerator first. There is likely a fan that blows air over the coil/grille and also directs over the frost-free evaporation pan.
Also look and feel if you have good air flow in the freezer section. Not blocking the flow with piled up TV dinners, etc. You can feel it. If it is hard to see the internal lights you might have too much in the freezer section. The freezer section usually siphons off air for the refrigerator section, too. So blocked freezer circulation could cause more air into the refrigerator section.
Hope this helps. All this you can do yourself before calling a repairman.
Good points..could also be that the door seals are degrading..letting humidity inside...they can be replaced fairly easily and inexpensively..