Sure. The rain can fall from above the fog and thus through it. Don't know about snow and fog. If it is "proper" snow, that is not sleet, it might be too cold for fog.
I regularly see rain + fog in HK
. As in I'm standing in fog and rain will fall on me.
At a guess, high humidity helps.
"There are no stupid questions, but there are a lot of inquisitive idiots." - John Ringo