Don Lively had a plan to bring more blacks and Hispanics into the practice of law.

Mr. Lively, a professor who is white, set out to open a law school that would take minority students even if they had low test scores or did poorly in college. Using retirement savings, a loan from his father and a check from a retired couple who read about him in a local newspaper, he opened Florida Coastal School of Law in 1996 in Jacksonville.

That...