Extremely biased here, due to my involvement with the team for six years, but I'm convinced it's 2791. And I'm not technically on the team right now so this post is OK
Here's a complete list of the close calls 2791 has had in their history:
BAE 2012: Second pick of the 7th alliance with 1519 and 885. Knocked out the 2nd and 3rd alliances on the way to the finals, won Finals 1... and died suddenly during Finals 2. Due to some confusion the alliance was not permitted a backup robot and they lost Finals 3 2v1.
CT 2012: Second pick of the #1 seed with 2168 and 118. 2168 was the best team to not win a regional that year, and 118 was an Einstein caliber robot. After cruising through the quarters, 118 encountered the same bug that disabled them on Einstein. Still, the alliance would have won Semifinal 2, except for the fact that after the buzzer, a robot rolled off the opponent's bridge, and the remaining robot perfectly balanced it. Blown out in Semis 3 and that was that.
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WPI 2013: 2791 ended up as the #1 seed of this event, picking the highest scoring robot and only floor loader of the event in team 20. In the quarterfinals, an arm gear broke, removing their ability to hang. After a frantic fix, the robot was still functional for semis and finals, scoring the highest points of the event until Finals 2, when 2168 discovered that if you hit 2791 just right, their arm now had enough slop in it with the repair to jostle their shooter into a jammed state. In Finals 3, a full court feeding strategy was attempted to mitigate this, which almost worked. Any of three separate events would have won the event - 2791's shooter working as they were hit into the pyramid by 2168 (changes a foul into a tech); 3182 not hanging in the center front of the pyramid on top of a disk (and us remembering the Q&A that clarified this is a legal hang); 20 slidng off the very edge of the pyramid at the zero second mark. This was the most painful close call I'd been a part of.
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FLR 2014: Less close than some of those above, but the #4 seed 341 declined the #2 seed to pick us and 4930. In the semi-finals, we ended up losing by a single missed truss shot in match 1. We were more easily beaten in match 2. Not particularly close but a strong performance.
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2015 never happened
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TVR 2016: 2791 is the first pick of the 3rd alliance, after some issues with the drivetrain and collector. Set some high scores quickly in the quarters, but were bested in the semis by 3 close matches after a substitution on the other alliance.
FLR 2016: 2791 is much improved, doing a solo capture at least once in quals. 2791 is the first overall pick of the event, by 5254. The second pick is a sleeper who was high performing at their last event but struggled early at FLR. After storming through the quarters, a combination of bad autonomous luck and shutdown defense by future Einstein finalist 1405 kept 2791 from even qualifying for Championship.
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So depending on how generous you're being, that's at least 3 events where a small change in circumstances would result in 2791 having won an event. And with the exception of 2015 (my bad), they have gotten more competitive every year since 2011, always running into a combination of mistakes, circumstances, and bad luck. I think they're overdue.