Mellon Institute of Industrial Research actually was originally founded as a department of the University of Pittsburgh (its first dedicated home is what is now Pitt’s Allen Hall) and the institute was spun off as an independent non-profit research institute that served as a defacto Pitt department until it merged with Carnegie Tech in 1967. The Mellon patriarchs (Thomas, Andrew, and Richard) were all Pitt alumni and major benefactors of the University of Pittsburgh. They had nothing to do with Andrew Carnegie’s technical schools during their lives. In fact, if naming conventions existed in the 1920s as they do today, the Cathedral of Learning would likely be named Mellon Hall.
Mellon Institute actually mostly did research that was contracted out by industry, not the government. It was spun off as a separate non-profit from Pitt and when companies began moving their research in-house after WWII, it started looking for a academic partner to merge with. There were several reasons why it didn’t merge with Pitt, including the fact Pitt was in the process of becoming a state-related university following a period of financial turmoil, along with, allegedly, an unwillingness of Pitt to incorporate Mellon into its name. In any case, Pitt still has several faculty with labs in the Mellon institute building which is located adjacent to Cathedral of Learning/Heinz Chapel grounds next to Pitt’s Bellefield Hall, as the two universities jointly sponsor a variety of institutes that find at least part of their homes there. Interestingly, in more recent years, there has been somewhat of a return to the trend of companies outsourcing their research.
Don’t feel too bad for Pitt. Pitt’s highly regarded and ranked School of Information Sciences was originally founded at Carnegie Tech. Such is the mixed and collaborative history of two universities that at one time attempted to merge and whose campuses literally blend together at points.