[UPDATE: See the end of the post for a possible solution, which works for some people.]
Many iPad 3 users are experiencing a bug which kills their 3G connection every time they connect to a Wi-Fi network. The cellular connection looks normal, with the carrier name, the 3G logo and reception bars, but when you try to do anything that requires a connection, it either times out or throws the error messages “Could not activate cellular data network.”
I am having the exact same problem, and here’s what’s happening.