I have a 2001 Savage (15K) that I currently use for a "winter beater" bike, which I typically just ride from Nov. thru Feb. here in NE Ohio. It will see 6-7000 miles during those months, and it gets ridden in all of the nasty weather. Twice this season I rode somewhere, and then the engine would not crank when it was time to go. At home, it is stored in an unheated barn, and always starts easily when I want to go somewhere.
First time, I rode it 12 mi, temp was -6F. It sat for maybe 5 minutes before I returned. Hitting the start button resulted only in a relay click. Headlights were bright, even when trying to start it. Pushed it inside the shop for an hour to warm it up, and it fired right up.
Second time, about 4 weeks later (and numerous successful sub -zero starts), I rode it about 2 miles, temp was -10F. It sat for about 3 hours, then again, only relay clicks. Tried to jump it from a truck - just clicks. Pushed it home.

I left for Guatamala the next day, so the bike sat for 2 weeks, with temps mostly 0 to -30F. The day of my return, it warmed up to 5. I gave the bike a try and again, it fired right up. It just sat the whole time, no battery charge or anything, so I would think that weather would have polished off a weak battery. That was back in late Jan, and I have ridden it over 1200 miles since then without incident.
Connections all looked good, but I cleaned them anyway. Bike is basically stock, but I have added a large windshield, Hippo Hands, and heated gloves for winter comfort. The gloves were not accidentally left on either time. I have had occasional trouble with the ignition switch being hard to turn because of salt and road grime, but a good cleaning takes care of that problem.
My only thought is maybe moisture freezes something up, or prevents a contact from being made? But then, I would think any temp below freezing should be a problem. I can't say as I want to troubleshoot a bike out in sub-zero temps, but it works fine otherwise. Hopefully we are done with the below zero stuff for this winter.
Anyone else ever have this problem? Any suggestions as to what the problem might be? I've ridden other Savages throughout the winter, and not had this problem with them. My apologies for the lengthy post.