Well can we start with Indian River State College in Florida??
Their Men's swimming team has 36 consecutive NJCAA national championships.
The University of North Carolina does not have any national championships in football.
Alabama and auburn
116 national championships since 1895 in football. However most teams don't count their national championship pre ww2!
UCLA has won the most consecutive NCAA Division I Men's basketball championships, but Kenyon Univ has the most consecutive championships in any sport with 30 in DIII Swimming and Diving. Hobart College has the most consecutive team championships in any sport with 12 in D III Lacrosse.