Actually, from what I understood from listing to a TTC lecture series about this, the "decline and fall" of the roman empire is somewhat of a fiction, in that it wasn't a brutal event, more of a long slow disintegration over the span of a thousand years. To give some sense of perspective, that's 3 times longer than the US has existed, and people have been clamouring that the US is in decline for the last 50 years.

What happened is that regions progressively became more independent and broke off from the central hegemony of Rome and then Byzantium. It took until the 1400s for that last official centre of Roman power to dwindle, and even through to today, Rome has maintained a huge influence on the rest of the world through the Roman Catholic Church.

So, rather than falling down, the Roman Empire slowly but surely transformed into something else. Arguably, we (western civilisation) are the direct descendants of the Greco-Roman Civilisation, and so one could make a (somewhat strained) argument that the Roman Empire is alive and well today, just in a different, more modern form, that has undergone 2000 years of evolution.

What to make of this myth of the downfall of the Roman Empire, the Dark Ages, etc? Well, according to that lecture series, this was largely a myth concocted during the Enlightenment to position it in history. The Renaissance needed to frame itself as a major cultural change, so it helped to frame history as "the Classical period, the Dark Ages, and now the Renaissance" - and draw on a fine period of history to support all the new ideas and revolutions brewing (i.e. "things were good back in Roman times, then things were really bad because of the Church, and now things are better again because we're standing up to the Church"). For the people living through the supposed transition between the Classical Period and the Dark Ages, though, apparently, there was no such stark difference.