In 2011, 2012 and 2013 the Sweet 16 of the NCAA men's Basketball tournament included Marquette, Kansas, Ohio State andFlorida. Therefore, there is no "only team" to achieve this distinction.
Through 2013, the longest streak of consecutive Sweet 16 appearances in the NCAA men's basketball tournament belongs to North Carolina, which advance to the Sweet 16 for 13 seasons 1981-1993.
From 1967-1980, UCLA was among the final 16 teams, but 1) its 1980 tournament appearance was vacated due to NCAA violations and 2) the tournament did not expand to 32 teams until 1975, effectively negating the concept of "making it to the Sweet 16."
The longest current "streak" belongs to Ohio State, which has been to the Sweet 16 for four straight years (2010-2013).
UCLA with 13!
Florida with 34
The current leader for most consecutive appearances is Arizona with 23.
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UCLA with 10, Cincinnati and Duke with 5
Eleven teams have played in consecutive Final Fours in the NCAA. UCLA has the record with 10 appearances from 1967 to 1976.
North Carolina holds the all-time record, with 27 consecutive NCAA Tournament appearances. The streak spanned from 1975-2001. With the inclusion of the 2009 NCAA Tournament, Arizona holds the longest current streak, with 25 consecutive appearances.
Including the 2008 NCAA Men's Basketball Tournament, the University of Kentucky has 17 consecutive appearances. Their current streak began with the 1992 tournament, and is currently the third longest active streak behind Arizona (24) and Kansas (18). Duke University is fourth on the list with 13 consecutive appearances.
As of 2009, 22 NCAA tournament appearances. 3 elite 8 appearances prior to the 2009 tournament.
The Penn State Nittany Lions - 87 consecutive match victories as of 10/30/2009 (and counting)
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Through 2013, Louisville has won three NCAA men's basketball titles (1980, 1986 and 2013). The school has made 10 trips to the Final Four and 20 to the Sweet 16. (The 20 appearances in the Sweet 16 excludes 1959 when there were only 15 teams invited to the tournament).