The 30-year-old was down the field going into the sixth and final discipline, the 160-lap points race, and was unable to improve his position as Colombia's Fernando Gaviria Rendon claimed a thrilling victory from charging German Roger Kluge.
Earlier Cavendish had finished a disappointing 11th in the 1km time trial before reviving his chances in the flying lap where he posted the fourth quickest time.
Cavendish made a good start to the points race but ultimately fell short of the medals, the target he was set by British coach Shane Sutton beforehand.
Track World Championships
Wiggins and Cavendish combine to win gold as Britain top medals table
06/03/2016 AT 18:00
The 26-times Tour de France stage winner has never won an Olympic medal of any colour, suffering disappointment in the road race in 2012 when he was one of the favourites.

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Cavendish is desperate to fill that glaring omission from his CV this year but his inclusion in the team is now in doubt.
Youngster Jonathan Dibben's brilliant victory in the points race on Friday further complicates the picture.
Dibben's surprise gold in his first world championships has given Sutton a possible selection dilemma as the 22-year-old is being tipped as an Olympic medal candidate in the omnium.
He is also a better team pursuit rider than Cavendish so selecting him would give Britain more options in that discipline in August when they are desperate to beat the Australians in Rio.
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Laura Trott clinches omnium gold for Great Britain
06/03/2016 AT 14:14
Track World Championships
Jason Kenny claims men's sprint world title for second time
05/03/2016 AT 22:53