It really depends. In Montana I'd expect there to be coyotes, coons, skunks, foxes, weasels...
Unless you buried your fence more than a foot or have some other impenetrable barrier impervious to patient diggers, I wouldn't risk it. If you're worried about not getting up in time to let them out, consider an automated setup for the door. I wouldn't risk it because of the time and money that have gone into my birds, but it comes down to doing what you want and learning the hard way if you take the easy way out by not cooping them up.
I'll add that I live in a suburban area and do not deal with predators outside of skunks and coons, and neither dig. I have not had a problem with them but I could, and I built my setup to accommodate that risk. If your setup is properly designed for your area they should be fine.