How much snow can cover a runway while still allowing commercial airliners to land safely? In regards to this question I thought about how much snow is shown on the runway and the exits in the movie.

So normally a commercial airport will have a fleet of snow plowers or heaters to keep runways, taxiways and aprons mostly clear of snow. But what is the safety limit on when the airport would have to close down due to too much snow coverage on the runway (not due to heavy snowfall, but due to the runway being covered completely in snow).

I think once the markers are no longer visible you would have a severe problem navigating on the ground or once snow piles up so high that the engines would take in a lot of snow it might also be a problem. But what are the actual limits during which one could safely land an aircraft on a snowy runway? And has it ever happend that someone landed on a runway covered in snow anyways?
There is no set legal limit (ultimately it is up to the PIC to decide) but condition reports are available to pilots at many airports to assist in assessing takeoff and landing performance.   The FAA has recently created a standardized way of reporting runway conditions through their TALPA (takeoff and landing performance assessment) initiative.

TALPA is an improved way of assessing runway conditions, based on contaminant type
and depth, which provides an aircraft operator effective means to anticipate airplane
braking performance. 

The RCAM (shown below) is the tool airport operators use to report a runway surface assessment when contaminants such as water, ice, snow, slush, and others are present on runways, taxiways, and aprons.   To find an updated RCAM look in Chapter 5 of Advisory Circular 150/5200-30, Airport Field
Condition Assessments and Winter Operation Safety.

The RwyCC is the runway condition code -- as you can see a code of 0 yields minimal to non-existant braking and/or directional control.