CHICAGO — Survey the schools that make up the modern Big Ten, and you find brand-name universities that actually have the football history to back up their bragging.

Michigan and its N.C.A.A.-leading 953 all-time wins. Nebraska, whose mention evokes memories of Tom Osborne’s bracingly powerful and speedy offenses. Ohio State, which has 37 Big Ten titles. Penn State and its prodigious stream of stirring linebackers. The league is awash in Rose Bowl victories and has amassed 51 national championships.

Yet it is not easy being part of the Big Ten these days, at least when it comes to reaching college football’s most consequential games. The last College Football Playoff did not include the Big Ten champion, Ohio State. The story was the same a year earlier. And the year before that, the conference champion for the 2016 season was left out while another Big Ten team surfaced in the elite quartet — only to lose, 31-0, in a semifinal.

That recent history forces the Big Ten — with its grand ambitions, glorious history and growing grudges — toward an uncomfortable question: How much will it matter who wins the conference in December?