In the Twitter case, there's nothing stopping the bully starting a new account and tweeting from that instead, post block. You might have complete control, but a lot of the time it's only after the fact.
There's no way for someone to know if they're blocked or not, though. It's not like Facebook where if you block someone, you are prohibited from interacting with them in any way. You can still send @messages, but the target won't see them. DMs can't be sent unless you're friends.
