The problem that I'm currently encountering is that the authentication server is down. I can start Minecraft and select "play offline". I then attempt to connect to a survival multiplayer server, and I receive the message "Failed to login: user not premium".
It depends on how the server is setup, but usually it is not possible to login when the authentication server on minecraft.net is down.
This is on purpose as most server administrators use this as a accountability method, ensuring the mods on their server are indeed the mods on their server, and enabling banning those that misbehave, even when they have changing IPs. (they would need to re-buy minecraft to connect again, if they're on another IP.)
For those servers that allow unauthenticated logins, you will be a generic "Player".