You know - when it's a cold day and your sinuses back up a bit and your nose runs a bit.
Horses absolutely do seem to suffer from the same effect. That is to say, I've definitely seen a bit of clear discharge in a horse's nostrils at the same time I'm reaching for a tissue myself. Given a runny nose in cold weather is partly caused by basic thermodynamics, it makes sense that horses get it too. (That said, if the discharge is thick, accompanied by a cough or, worse, only in one nostril, then it might well be a sick horse). Another clear sign that it's just the weather: When it's every horse in the herd.