My initial thought was it would probably be somewhere in the ballpark of 20-25% of the time. Often when a team makes 22 free throws, it's because they are winning and the other team is fouling late in the game.
I looked briefly at Minnesota's games over the past several years and found that a team making 22 or more free throws in a game (either Minnesota or their opponent) has had the following record:
2011-12: 2-1 (2-2 for 21+ makes, as Minnesota lost to Dayton despite making 21 FTs; in that game, MN had an eFG of 51.0% vs Dayton's 56.7%)
2010-11: 6-1
2009-10: 6-1
2008-09: 1-2
More important than the largely irrelevant free throw percentage (though not very important to winning a basketball game) is just getting to the line. In the Big Ten this year, there have been 22 games in which a team has shot 75.0% or better from the line.. those teams with good FT% performances are 10-12 in those games.
If you think of a 60% free throw shooter (excluding the small impact of front end one and ones), you'll hear a lot of complaining about their FT%... but, when they get to the line, the team is scoring 1.20 per possession. You'll win a lot of basketball games if you're scoring at a 1.20 ppp clip.
