Not really. Microsoft says you can play offline BC if you've set your Xbox to your main console, but this doesn't seem to be true in all cases. We have the one Xbox One. It's set as my main Xbox. When our internet goes down, the Xbox basically turns into a brick. We can watch blu-rays and play some games, but not all, and definitely none of the BC titles as far as I've seen.
The full offline support page lists some caveats:
I'm assuming digital BC games fall under here.