You don't say wether or not your model is a side by side or over-under, which would be helpful to know.
Problem is, if you're hearing a "hum" you should also be getting airflow through the unit. I'm assuming an American style that has a single compressor unit that provides cooling for both the freezer and the fridge. In fact the cold air intially comes into the freezer section and then, through a motorized air flow control door (which is what the thermostat actually controls) between the fridge and the freezer.
Try this... when the "hum" is occuring, open the freezer side and see if the fan (should be at the top center, is working. If that's working place your hand over the exit of that doorway in the fridge side (should be at the top of the division wall, if a side-by-side) to see if there's any air flow entering the fridge. If the fan's working but no flow to the fridge side the control for the flow control door is inopeartive. This can be replaced by a reasonably astute DIY'er. You'll need the model and serial number to get the correct replacement part.
If there's no airflow anyplace, slide the unit out and, while the humming is heard see if any of the circulation tubing (usually copper piping) can be reached. Careful! Feel it to see if it's hot. This is also an indiaction the compressor is working, If that's the case, I'd be additionally suspicious of the fan itself, which can also be removed and repalced by a good handyman...
At any rate, Best of Luck!
Addendum: I realize I'm making judgements on my experience with units here in the U.S., so if there are differences,my apologies...