How does US healthcare support the mentally ill?

Given the private nature of US healthcare, how much of a role does the state play in dealing with those who are mentally incapacitated?

Say as an example, a family who has a seriously mentally retarded child, and as the child grows older it becomes clear that they would be better looked after in a dedicated care institution. Is this a matter for health insurance, like any physical illness? Or does the state play a role in contributing support for the mentally ill?