In the case of electrical storms, lots of people can feel the electrical tension in the air leading up to the storm, and also the increased humidity, too. In the case of non-electric rain or snow storms, not so many people can sense those coming, but there's still a good many who can. I can often smell rain in the clouds overhead before it starts to rain, and I can almost always smell the snow in the clouds before it starts to fall. What's really funny about it in my case is that I have a poor sense of smell for many things, but I can still smell some things, including the weather, which strikes me as odd, because water doesn't have a strong smell. On the other hand, I can't smell skunk, and yes, I've had the chance to smell skunk every time I've driven past dead ones, and the two times one of my dogs had run-ins with skunks. I was the only one in the family who could stand to be near him until after his tomato juice baths.
Anyway, I think this weather sense is kind of built into us, but in some people it's either missing, or defective, or just not used.
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If a man does not keep pace with his companions, perhaps it is because he hears a different drummer.
Let him step to the music which he hears, however measured, or far away.--Henry David Thoreau