I don't know about the wood, but the pressure from the snow being piled there might stress the posts.
An example:
4-5 years ago this area experienced a tornado. We didn't experience any damage, or so we thought. A month later there was a string, about 30' high, that went across our yard, held up by the trees. We were out looking at it and noticed one of the fence post on the west side (wind went from west to east) was leaning in a little.
After examining:
26 years ago, when we put the fence (wood, maybe cedar but don't know for sure, posts are pressure treated) in, we put a 1/2 bag of concrete in each hole, then tamped down the dirt on top of it, filling the hole. The holes were 30" deep. We didn't wet the concrete, the rain did that for us - saved us a step.
The post that was leaning in a little had broken off where the concrete met the dirt above it - the fence attached was holding it up.
The wind was not tornado force, but was strong enough to snap the pole.
The fence weathered to a nice gray. There are parts in the back that experience 3-4' snow and look no different than that which doesn't