We all have an open NAT most of the time and we still experience this problem. We also use wired connections. I thought it was the NAT as well, and then I tried experimenting with my router to see if it would work. Another interesting thing came up yesterday, though. I was able to communicate with an entire lobby (apart from the people who I labelled as Friend and Friend) and they were able to as well.
It simply doesn't make sense. Sometimes we can communicate perfectly fine, and then other times we will have to wait until we're in pre-game lobbies or actual matches. It's frustrating.