I've heard of people feeling a little head cold the next day, but never with a stiff neck. In any case, both claims appear to be old wives tales. I'll keep a fan running from spring to fall (I like the white noise in the background) and have never had any discomfort issues come morning. Perhaps you are correlating the wrong things, and should look at your sleep position: how you fall asleep, how you wake up, the number of pillows you're using, etc. Any fixed position held for an extended amount of time isn't good for muscles or joints. Try switching things up a bit each night to see if symptoms recede.