Same thing happened to me with all my DS and 3DS games (also while being able to connect to the store and web browser). And on Xbox Live I could only connect with certain users, even excluding some from my own country. Sad thing is the problem wasn't solved until I got a new router with faster connection (jumped from ~54Mbit/s to 72) than the previous one. I looked up some solutions long ago, but all I could find was some complex solutions that I either failed or never bothered trying out...
Sorry I can't be of any help.







