I have no connection issues, but it appears the galaxy at war status servers are very badly overloaded. When first starting ME3 (PC) it almost always claims galaxy at war can't connect to server, but I can still authorize DLC and go to multiplayer. Sometimes getting to multiplayer gives it time to update galaxy at war, but more often than not it now (since a few days ago) fails here too. Sometimes going in and out of the store give it time to finally connect, but now it does so less often. Sometimes actually purchasing a pack at the store will fix it when going in and out of the store won't. Now it mostly doesn't.
Just to emphasize, I connect and play multiplayer fine. I can go to the store and buy packs just fine. There is no issue at my end, and the servers for galaxy at war are entirely at fault and do not respond in time. Many other players have mentioned the same thing to me. All of this seems to have started sometime in December, and is steadily getting worse. I don't know how badly this will affect the game, though I suspect it only hurts single player. I'd like to know when the servers for galaxy at war status will be repaired though, as part of the system is now obviously failing and getting worse daily.
