The internet has destroyed that. IMHO, it is the single worst thing the internet has done.
I sincerely hope no one suggests that parents should exclude their kids from the internet which allows the bullys to win and control cyber space. Worse still, it then excludes them from the friends they might still have, and a lot of things the victim can use to fight back, and distract them selves from the nastiness. The internet is more good than bad.
I think laws for cyber bullying should have harsher penalties than normal face to face bullying. Face to face stuff is limited by physical proximity and time, cyber bullying can be 24/7. If bullied at school, the kid can escape it by going home after school, and not being there at weekends. With cyber bullying, you might not even get an evening off, let alone a week end.