Both of mine were in a similar situation. I worked at a private school, which allowed both of my children to go there for free. DD4 was in a pre-k program for 3year olds, and DS7 was in kindergarten. This year we moved to another state. Here we have no private schools, and the pre-k programs are, quite honestly, a joke. DS7 was put into the gifted program, but still bored to tears. The problem with public schools is that, partly due to No Child Left Behind, they are forced to cater to those kids who are struggling, often at the expense of those who are not. Private schools don't have to do that. Cruel as it sounds, if someone in a private school can't keep up, either their parents get them a tutor, or they go back to public schools. Now, with that having been said, if you get a good teacher in a public school, it helps a lot. DS7's teacher has done everything she can (within the limits of the given curriculum) to challenge him. But I still supplement at home and keep a notebook in his backpack full of extra stuff for him to do.
So in short, I firmly believe that any child can get a good education at any school - but sometimes it takes a heck of a lot of work on the part of the parents.
