Are you getting disconnected from Live itself, or just getting kicked from games?
There's currently a routing issue that started happening last night. Every few minutes, certain servers/sites become unreachable for a short period of time, then it'll start working again, but obviously if you're in an online game when that happens, you'll get disconnected from it.
I know Overwatch is affected as I was kicked from it multiple times last night and I've been kicked 4 times so far in the last 90 minutes tonight. When I get kicked, I'm still connected to Live though as I can still chat to friends in a party.