Yep. The Crisis of the Third Century, especially.
But also, pre-imperial Rome, a number of times. The Conflicts of the Orders, the entire late-Republican period from the Gracchi to the final demise of the Republic under Augustus.
And then the Eastern Empire had to reinvent itself multiple times. The near-collapse to the Caliphate. The Norman invasions of the Balkans and the First Crusade. The Fourth Crusade and its aftermath.
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Then in the West, you have a couple brief periods where Rome isn't subject to Eastern Imperial overlordship. Justinian fully reconquers it in the late 500s, and Rome stays nominally part of the Eastern Empire until its brief loss to the Lombards in the 750s. But Charlemange's father, Pepin the Short, reconquered Rome and confirmed the Pope's authority over it. A couple decades later, Charlemagne is crowned Emperor of the Romans. And Rome exists under a constantly shifting balance of authority between Popes and Emperors until the Italian Unification in the late 1800s.
It should also be noted that even before the end of the Western Imperial line in 476, Rome rarely served as the actual home of the emperors since the late 200s.
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So there's an interesting question of whether Rome ever did really fall, before the modern period. I would say "yes", in the sense that after the mid-400s, you never again have a sense of authority over the Mediterranean basin eminating from Rome (or the northern half of Italy). That's what people think of when they think of Rome. There are only brief periods of anything like that, and you never again have multi-generational, institutional authority.
But in other ways, there really is a continual reinvention of political systems that trace their lineage and cultural power to Rome. In that sense, it's kind of like China, which is similarly not really a continuous institution.