Another thing to check when your furnace is doing that is the outside vent from the furnace. If it get's plugged with snow or ice the furnace won't vent and it won't allow furnace to start up.
Yes, this happens when it's really cold. The condensate from the furnace exhaust will freeze where it comes out the side of your house. And eventually block the exhaust. So the furnace will stop as a safety measure to keep CO from accumulating in your house and killing you.
My brother had this happen when he was in Florida about 10 years ago. Super cold snap then it warmed up. When he got home he opened his garage door and a wave of water came out. Furnace had stopped and the pipes froze. Then when it warmed up the ice accumulation on the exhaust melted and the furnace started working again.
Thank god his insurance covered most of it but it was a huge mess and he lost a few personal items that hurt.