On the 18th of April, I along with the rest of my family members all started having the "FaceTime connection issue" around the same time and day. It only seemed to effect our iPhones. As we could FT ipad to ipad or iPod to ipad and vice versa with no problem... Our phones where on the iOS 6.1.3...
I opened a support chat with an apple tech and they couldn't tell me why this happening. However, they told me that I would need to upgrade iOS.... I was really trying to hold off upgrading, but wanted the FaceTime feature to work. So, after trying everything possible to make FaceTime work without upgrading and having no success, I upgraded the device. I tried FaceTime and it works fine.... I think Apple was doing some kind of maintenance on the 17th, after that many people (globally) started having issues with FaceTime.... Go figure... "I think it was a Conspiracy, to force everyone to be on the new iOS" Anyway, Upgrade it, you will feel like you have a new device! I'm glad I did!