I suspect it will know where the lane markings are better than human drivers. They are mapped ahead of time and the car can likely localize itself via other landmarks to determine where they are without being able to see them.
The harder part is driving like a human and detecting that a path has been made in the middle of two lanes in heavy snow and not obeying the lines at all.
The first idea seems like it would require a lot a lot of data stored in the car. Is it feasible? And even so, to be that dependent on matching up with existing pre-mapped data suggests a system that would be quite slow to roll out across a country.
The harder part is driving like a human and detecting that a path has been made in the middle of two lanes in heavy snow and not obeying the lines at all.