My parents were both private-school and postgraduate level educated, and worked in academia, but were very anti-private school, so I went to an ordinary comprehensive (sat the 11+ but didn't pass it).
And yes, from day 1 at secondary school I was bullied for being posh, for being clever, for being boring, for being weird, and I hated every minute of school.
Academically it wasn't a bad school and the teachers were in general very good, but they seemed unable/incapable of doing anything to support more than averagely bright pupils - I was used to being told 'I know YOU can do this, guinealady' or 'If you've finished already guinealady you can sit quietly and get on with something else' - no-one tried to stretch me at all or make me feel I was anything other than an inconvenience for being bright. Added to that they all turned a blind eye, as far as I could tell, to the bullying and it was a miserable experience all round.
However my dad was bullied at private school too, and I don't think for a minute the only reason I was bullied was because I was posh - I would still probably have been picked on for being dorky and weird at a private school or grammar. I must have just had one of those faces that said 'I'm vulnerable and shy, pick on me'.
I switched to a grammar school for my A levels and was much happier there - it was much more academic and there was so much more support from the teachers - but I'd also put that down to it being a smaller school and smaller class sizes in the 6th form means you get more one-on-one support.
I'd love to say that being bullied toughened me up for life and I'm a better person because of it, but if I had the choice, I would rather not have been bullied.
We will probably move out of London when the time comes, a prospect I'm already half-dreading, but I just can't bear the thought of a child of mine going through what I had to.