Worst NFL team vs. National Championship runner up

Question

Were there ever games at the end of football season pitting the NFL team with the worst record against the runner up for the National Championship? Keep in mind, I'm not asking who would win because I would imagine the college team would get slaughtered.

Answer 1

They used to actually have the NFL Champion play an All-Star team from the college ranks, called it the College All-Star game.  The college teams actually had some success, winning 5 of the first 14 contests.  After that, the pros dominated and the game was discontinued  in 1976

Answer 2

Back in the old days, 30's and 40's, the NFL champion would play the college all-stars.  It hasnt happened any other time that I remember.  

The 2001 Miami Hurricanes probably could have beaten the worst team in the NFL that year, Cincinnati.  

Answer 3

Worst NFL team.

against the runner up or, the national champions.