I'm gonna make this harder on myself, and go with only teams that did not win the national title. Obviously.....2002 Miami is going to be excluded from this list since they did in fact win....
1. 1988 - this team was truly great, ruined FSU in the season opener and engineered an amazing comeback in front of a 6-figure crowd at the Big House. IMO, would have easily won the national title, but of course, were given the business from the infamous zebras in South Bend, IN.
2. 1990 - a 2-loss team that for whatever reason just didn't show up to play BYU in the opener (I think their conditioning at that time was just not up to playing at Provo, UT elevation), and lost a tough game at, again, South Bend. Obviously, they went on a run after that game, destroying teams, ultimately peaking at the end of the year and destroying Texas in the Cotton Bowl. If there had been a CFP at the time, this team would have been a problem, and may well have won another title.
3. 2025 - no need to go into any detail, we all just lived it.
4. 2000 - another team that dropped a game early in the season, but peaked at the end, and very, very likely would have beaten Oklahoma for the national title had the BCS not put FSU in instead.
5. 2017 - this was that team that really gave you a glimpse of what Miami being good in the modern era could look like. I still think this team could have gone further than it did. Rosier fought his *** off for the squad and is a true Cane in every sense, but just didn't have the arm that team really needed.
Honorable mention - 2024, 2020, 2016, 1999, 1998, 1994, 1981. I'm leaving 1986 off because I honestly still don't know why they lost to that Penn St team. And 1992, the manner of their loss against Alabama was just embarrassing, I was there. It was almost as if they just expected to walk on to the field and beat a very motivated team.
If I had to pick one team, it'd be the 1991 team. Man, that team just did not give up any points.