Growing up, I was taught that the Roman Empire was split in two by diocletian, East and West, and that the West "fell" in the 5th century.
But as always, the story is far more interesting than that.
The split was administrative only -- meaning the Empire remained intact. There were augusti over East and West -- but there were also prefectures, diocese, and provinces. Diocletian created a federal system for a united Empire.
He also moved the capitol of the Western portion to Milan -- so Rome was no longer the capitol of ANYTHING significant.
But then Constantinople became the de facto heart of the Empire -- because of its fantastic trade location, that's where the money went. And because that's where the money was, that's where the power went. And the West was neglected, and became something of a backwater -- the center of power lay in Constantinople, and the rest of the empire suffered a brain drain, as always happens in every highly centralized state.
And eventually, the Germanic tribes from up north took the Western provinces -- but this was a loss of territory to a single Roman Empire now effectively capitoled in Constantinople ... not a "fall of the Roman Empire."
That event -- the true fall of the Roman Empire -- occurred in 1453 when Constantinople was conquered by the Ottomon Turks.
If I were to guess, I think our current view of the Roman Empire stems from:
a) cultural bias on the part of Western European historians that see only the Western portion of the empire as significant;
b) professional bias on the part of Western European historians -- because which makes a more interesting story -- "The Ancient Roman Empire Crumbled from within into a ball of fury from mysterious causes which I will now elucidate" or "Nobody cared about Rome anymore, because all the money was in Constantinople -- which is studied by the professor down the hall."