Which means the degree that they conflict with the character will determine how much they drive the plot.
Picking a harmoniously simple background for a character -- the wizard whose parents were wizards, the cleric whose devotion led to dedication -- makes that background fade into the background. Which, obviously, is often the right thing. Having all the characters with quirky and attention getting background makes the story garish.
Picking a conflicting background is an aid in deepening the character. Having a character harmonize two apparent contradictions makes for round characters, even something so simple as having a character be lazy and industrious at different times.
A background that doesn't fit has a whole range. If a rogue has a background as an noble, it can go from the simplest -- a decadent set of nobles where it's regarded as suitable -- through adolescent rebellion, up to mastering it for purposes of opening ancient treasure chests after the keys had been lost through folly. All of which would have plot consequences.