Do you have icicles hanging from the gutters? If so, then heat leaks from the house to the attic and causes the snow to melt. When ice dam forms at the lower edge of the roof water is trapped and goes up the shingles. The water can run down the vertical walls and go to the ceiling of the first floor. This is common for roofs with low pitches. The solution to this problem is: 1) improve insulation in the attic; 2) install vents at the eaves.
If the above doesn't apply, then you just have a leaky roof!
If the above doesn't apply, then you just have a leaky roof!