404 on nested custom post types

The problem, from the viewpoint of the WP parser, is that it doesn’t know if you want the school-name post or the teacher-name post. It would rather return nothing than the wrong post.

You can help it understand what to do with the Rewrite API.

Another thing is, what is a parent of what is not implicit in a post record. Have you considered using a single hierarchical CPT for the entire structure? Say an “education” CPT. All top level posts are schools (further defined by a post meta field). All schools have teacher children. All teacher posts have exercise children.

Just a thought, maybe it’s too late for that. It results in a different permalink structure which may or may not work for you. You could try making some quick test pages (page post_type) to see how such a hierarchical structure would work.