Its hard to defend your borders when you have to worry about a civil war. Thats why they split time in two and doubled the army.
At least thats how I see it
Rome was still there when most thought it fell, only it was called Byzantium.
West Rome fell because they could not find adequate rulers anymore, the Republic ended in civil war (mostly fueled by personal egotism --- wait where are we seeing that today...hmmm?) and the Empire because the hereditary system only works until inbreeding gets the better of it.
East Rome because they could not defend themselves against Ottoman cannons with ballistas.