I would only consider sending any children I may have to a private school if I felt in that specific situation that it would be better for them. I wouldn't want it to make arrogant or anything like that, but if it provided them with a better standard of education then yes I would.

I agree with Jim though, if a child is intelligent then they can do well in most schools, and I think it can also be down to what they do at home, as home schooling could be a good option too, albeit that could leave them with a lack of good social skills.