You raise an excellent point and one that goes to the heart of what can be wrong with schools. In short, the policies and practices of schools often work to put students in competition with each other and don't work to foster a sense of shared community and common regard and sadly your children are feeling the brunt of that.

Schools have traditionally only valued a very few skills - memorization and those who are good at it are often held in high regard by schools. The ability to sit still for long periods and to be quietly and consistently compliant to school rules are two other qualities that schools hold in high regard. So that students who may not read as well or as quickly as others, students who don't memorize well, or sit still well often become 'second-class citizens' in school. Teachers tend to shower the most praise on the children who seem to need the least teaching. It leads some children to believe that they can do wrong and that other children are very wrong not to be as good as they've been told they are.
There is a popular bumper sticker around here that says "My child is an honor student at _____ School and every school seems to have their own version of that same bumper sticker. Yet one school has put out this one -
"At Green Street School, every child is honored."
And in the better world, every student would be helped to find their own particular gifts and would feel honored and all students would be taught to treat each other with equal regard.
While we still live in the world as it is, I don't know if it helps to know that heaping praise on some students is a mistaken but common way of trying to manage students. The thinking has been that the other students will shrink from shame and either occupy themselves with 'trying harder' or quietly accept their second class citizen status. In either case, it's been believed that they'll ask little of teachers who can then continue to heap praise on the students who need the least teaching.
Yet every child in the school sees this played out again and again and it's a great temptation to any child to gloat when they've been found to be one of a favored few. Some schools and some teachers are better at building a sense of community but those schools that have special programs for particularly talented students have then a particular challenge to fostering a sense of common regard. There's been a slowly rising school of thought that says such programs do more harm than good.
One of my own sons attended a school with such a program and it was much the same there.