As of 2010 this has never happened. A number one seed has never lost to a 16 seed.
Villanova, North Carolina State
The lowest seed to ever win a NCAA Basketball title was Villanova, ranked seed #8. They won the tournament in 1985. Lower seeds have made it to the Final Four (George Mason and Louisiana State, ranked 11th, made it in 2006 and 1986 respectively) but never won. UCLA in 1980 was ranked 48th and made it to the championship game.
Arkansas beat Duke, 76-72, in the 1994 NCAA Men's Basketball Championship game.
ccny
Princeton hasn't won a game in the NCAA tournament since 1998 when they beat UNLV in the first round, 69-57.
Kentucky
Michigan was a #3 seed when they won the NCAA Men's Basketball Tournament in 1989.
Michigan has won 1 NCAA Men's Basketball Championship, in 1989.
They have never won the tournament. They have only appeared in 3 NCAA tournaments and have never won a game.
In 1998 the Kansas University (KU) Jayhawks were the national NCAA Men's Basketball champions. In that game they beat Oklahoma 83-79.
UCLA is the NCAA school that won seven basketball championships in a row from 1967 to 1973. As of September 2014, UCLA has won a total of 11 NCAA basketball championships.
UCLA