Yes being near a cold draft or sleeping or sitting near a cold draft can lead to and cause a stiff neck.
I get a stiff neck when I'm near a cold draft such as when I'm sitting on my couch and my drafty windows let in some cold air.
I have to put a blanket over that window especially when the wind is blowing to keep the cold draft out and then my stiff neck goes away.
Sleeping in a wrong position or straining your neck can also cause a stiff neck as well.
Sleeping on your side with one side too high or low can stretch or constrict the neck muscles that can lead to a stiff neck as well.