The worst ever example of that was BYU during their MNC year.  In their very first game they played a highly ranked Pitt team and that helped catapult BYU up the rankings.  The thing is that Pitt sucked in 1984 and finished 3-7-1.  
It gets worse.  BYU only beat Pitt by six points.  At the time the margin didn't matter.  Pitt was highly ranked so the win was a big deal and BYU moved from unranked to #13 while Pitt dropped from #3 to #17 (back then only 20 teams were ranked).  
Pitt lost that year:
 - by 1 to Temple
 - by 6 to BYU
 - by 6 to Syracuse
 - by 18 to WVU
 - by 20 to Miami, FL
 - by 24 to USCe
 - by 32 to Oklahoma
In retrospect, BYU's six point win over Pitt was nothing special at all.  
I'm going to expand on this because it is a pet-peeve of mine.  Per the 1984 Final Poll, BYU played exactly ZERO ranked teams.  The closest they came was a five point win over Air Force.  Air Force finished fourth among "others receiving votes" so they would theoretically have been ranked #24 if the poll had been a top-25 (not necessarily because other teams might have gotten more 21-25 votes).  
Among teams that were ranked in at least one 1984 poll, BYU played:
- Michigan was ranked in 6 polls in 1984, getting as high as #3 and BYU beat them 24-17 in the Sun Bowl.  
- Pittsburgh was ranked in 3 polls in 1984, getting as high as #3 and BYU beat them 20-14 in their opener.  
That is it.  
Pitt:  The Panthers were ranked #3 in the 1984 preseason poll.  They were ranked #17 in the 9/4 poll after losing to BYU (2nd poll).  Then the Panthers had a bye week and they were still #17 in the 9/11 poll (3rd poll).  The following week the Panthers lost to Temple and they were not ranked again in 1984.  
Michigan:  The Wolverines were ranked #14 in the 1984 preseason poll.  They didn't play Labor Day weekend and they were still #14 in the 9/4 poll (2nd poll).  Then they beat #1 Miami, FL (who ended up 8-5) and moved up to #3 in the 9/11 poll (3rd poll).  They lost to Washington on 9/15 and dropped to #16 in the 9/18 poll (4th poll).  They beat Wisconsin on 9/22 and Indiana on 9/29 and moved up to #14 and #13 respectively in the 9/25 and 10/2 polls (5th and 6th polls).  Michigan lost by 12 at home to Michigan State on 10/6 and they were not ranked again that year on their way to a 6-6 record.  
The bottom line is that BYU played the most ridiculously weak schedule ever for an alleged National Champion.