Cellular data shutting off when i shut off wifi and vice versa
I have a Nexus 5 with 5.1.1, and was experiencing the same thing.

It was a bit weird; I think the problem started with the 5.1.1 upgrade, but it's didn't kick in fully as a problem 100% of the time until sometime last week (1st week of September). Prior to that it was an intermittent problem, dropping about quite regularly, but returning pretty quickly. 

Anyway, I rooted the phone, and reinstalled 5.1 and it's all working properly now.