We have two of the small bushnell cams and they have worked for weeks in 30 deg weather. As said before if the battery is a little low to begin with, the cold will drain them and make the cam shut off. Once the batteries warm back up the camera works again for a little while.
We have had this happen with cuddebacks, bushnells, and moultrie. I think the bushnells are one of the best cams out there, especially with all the features you get for $199.