When the NCAA tournament begins, do you ever stop yourself and wonder, “Hmmm, has that school ever made the Final Four or not?”

A few weeks ago, Run The Floor delved into the debate concerning the most accomplished programs that have never made the Final Four. In this piece, we’ll take a slightly different tack: What are the 10 most prominent schools in this tournament — the big names, not the North Carolina Centrals, Woffords, or Stephen F. Austins — that have never reached college basketball’s Promised Land? You might be surprised to find certain names on… and off… this list.

10 – TULSA

The Golden Hurricane now reside in a weakened version of Conference USA (one that no longer has Memphis), so a trip to the Final Four anytime soon seems unlikely. Yet, this school — which made the Sweet 16 under Tubby Smith in the 1990s and was a highly-seeded force (a 3 seed) under Nolan Richardson a decade earlier — never did break through when it had a good chance to do so. Tulsa-Wichita State was one of college basketball’s hottest rivalries in the 1980s; the Golden Hurricane should not be forgotten in a survey of high-profile programs without Final Four appearances.

9 – NEW MEXICO 

The Lobos have developed one of the great home-court advantages in college basketball over time. Their enduring excellence within the Mountain West Conference has lifted them to a No. 3 seed (2010) in the NCAA tournament. The Lobos have gained protected (top-4) seeds on other occasions. Yet, they haven’t been able to make the Final Four. Moreover, they haven’t been able to get to the Sweet 16 since 1974.

8 – SAN DIEGO STATE 

The Aztecs’ history might not be a long one, but this program has become so strong in the past few years that the lack of a Final Four appearance will become more frustrating as time goes by. San Diego State was a No. 2 seed in 2011 but could not get past the Sweet 16. In many ways, the failure of the Mountain West Conference to get a single team to the Final Four (or even the Elite Eight) is the reality that overshadows both San Diego State and New Mexico in each new NCAA tournament.

7 – ARIZONA STATE

The Sun Devils made multiple Elite Eight appearances under their greatest coach, Ned Wulk. In the 1963 NCAA Tournament, the Sun Devils won a Sweet 16 game against a man named John Wooden. Yes, that was UCLA’s last loss in the Big Dance before its run of 10 championships in 12 seasons. However, ASU lost to Oregon State in the Elite Eight 51 years ago. It has never been able to get over the hump.

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6 – TENNESSEE

This man (above) won’t be able to guide the Vols to their first Final Four. He’s taken, as of yesterday.

5 – BYU

The Cougars can’t be expected to make the Final Four in this tournament, but they’ve made almost 30 NCAA tournament appearances. One day, maybe they’ll be able to cross the threshold.

4 – NEBRASKA

The Cornhuskers have not only failed to make the Final Four; they’ve failed to win a single NCAA tournament game. Tim Miles is set on changing these and other realities in Lincoln. He might not do so this year, but given time over the next eight to ten seasons, he just might break through, given his coaching acumen.

3 – SAINT LOUIS

The Billikens have established a strong basketball brand for most of the past 20 years. As they solidify their place in the sport, perhaps they’ll find the team with the offensive firepower needed to reach the next level.

2 – GONZAGA

The pieces seemed to be in place last year, but Wichita State decided to shoot the lights out from three-point range, which was not one of the 2013 Shockers’ consistent strengths. If Mark Few can reel in another top-tier center and surround him with knockdown shooters, Gonzaga could create the kind of team that can make a deep run in March. One day, one way or another, this program hopes to reward its fans with college basketball’s most central achievement. The national championship is a truly rare prize; the Final Four is the glittering and more accessible goal.

1 – CREIGHTON

The Bluejays are, simply stated, the highest-seeded team in this tournament without a Final Four appearance. If Doug McDermott really wants to add to his legend — not to mention the historical resonance of his Sports Illustrated cover appearance — he’ll pull a Larry Bird and bring a former Missouri Valley Conference school to the Final Four.

BONUS SECTION:

SCHOOLS THAT HAVE MADE THE FINAL FOUR… BELIEVE IT OR NOT

The Wichita State Shockers (last year, but previously not since 1965). The Kansas State Wildcats (1964). The Saint Joseph’s Hawks (1961). The Colorado Buffaloes (1955). The Baylor Bears (1950). The Iowa State Cyclones (1944). The Pittsburgh Panthers (1941). The Oregon Ducks (the first NCAA tournament back in 1939). Each of these 2014 tournament teams has managed to be one of the four teams left standing at the end of a college basketball season.

Colorado has made a Final Four and BYU hasn’t?

Baylor has reached a Final Four while Creighton hasn’t?

So many starving programs are yearning to find the nourishment they’ve so desperately pursued. We’ll see if any of them can finally break through this March.