I have an iPhone 6 with iOS 9.2. I've recently frequently found that incoming calls frequently ring briefly and then the ring tone ends and I get an "Call using iphone failed" error on the screen. I do not have this problem with outgoing calls.
I've searched for this error and found a number of threads, but not with any clear suggestions or solutions.
I noticed that I only had the problem when my iPhone was near my MacBook pro. It turned out that the problem had to do with the Handoff featured introduced in iOS 8 and OS X Yosemite.
When I turned off Handoff and Facetime integration on my iphone, the problem went away. Here are a couple of threads with more details on how to deal with this:
https://discussions.apple.com/thread/6538204?start=0&tstart=0 (final post in thread)
And here's a possible fix that I haven't tried:
