It's fine, and I quite like Rob or Robbie. Not sure if we're ready for Bob yet (can hear Rowan Atkinson as Blackadder pronouncing it!), Bobby would be OK I suppose, maybe a little child-like.
BTW of course it's 'a classic like John or William'. It's a name that's been in use in Britain for hundreds of years. That's what a classic is. Heard of Robert the Bruce? Paul is a classic too obviously. Saint Paul? Even Alan really although it's out of fashion. One of Robin Hood's gang was Alan. Just because a name's become less popular for a while doesn't stop it from being classic.