Depending on what Zakalwhe meant by “State University” (considering his distinction between “State Name STATE” as the land-grant college and “University of STATE” as the liberal arts college), would Auburn and Clemson qualify? I don’t know enough about Rutgers to compare them, or does the existence of U of Ala. and USC rule them out?
Also, on the OP, I would add that universities named after places would generally take the “University of _" form and universities named after anything else (such as major donors) could take the " University” form. Yale University is named after Elihu Yale, and the University of Connecticut is in Connecticut.
I think there are probably infinite examples of this, actually. Not every university can be the top school in every field. For example, while Penn State may have a fine teacher education department, the education you’ll get at Shippensburg or any of the other PA State System schools may well be better.
FWIW, the Universities of Texas and Tennessee reached an agreement in 1986 over the use of “UT.” Tennessee had applied to the federal government for the right to use the trademark on a number of items not covered under Texas’ trademark rights.
Tennessee uses the trademark east of the Mississippi River and in Louisiana’s West Baton Rouge Parish, home of Southeastern Conference member Louisiana State University. Texas owns the trademark everywhere else west of the Mississippi.
In addition, the agreement gives the Longhorns and Volunteers limited rights to use the interlocking UT logo design in each other’s territory during bowl games or other special events.
In my experience, the use of [Statename] to mean The Flagship of the State University System is primarily in a sports context. “Michigan” = U of M, Ann Arbor. “Texas” = U of T, Austin. “California” = U of C, Berkeley. But in an academic context, I doubt that somebody would say they went to “Texas” …they’d be more likely to say U of T.
And then, Berkeley is a special case. I, and most alums, say “I went to Berkeley*”. They’d only say “I went to Cal” in a sports context, with “Cal” being a universally recognized abbreviation, like Ole Miss or Mizzou.
*Through gritted teeth. The University and the City loathe each other, and the city of Berkeley does everything in its power to impede everything the University attempts, especially athletically.