To a large degree Rome fell victim to the same dilemma that all empires that existed before the invention of rapid communication and transportation did. Without the ability to communicate with and quickly respond to any situations in the outlying regions of the empire, emperors had two choices, both unsatisfactory. Either they had to trust the local leaders with a fair amount of autonomy to react to events as they saw fit or they had to rein in those leaders. In the first case, the result is loyalty to the local leader rather than the far away emperor. This leads to destabilization from within. Emperors become paranoid and focused on defending the throne from their own subordinates rather than actually governing the empire. In the second case, when the emperor¬ís representative has little power, it becomes a situation conducive to rebellion among the provincial population.
Therefore we probably should not wonder why Rome did not last longer, but rather the more interesting question: how did Rome manage to last as long as it did. IMO they managed to strike a good balance between the two options above, and often granted Roman citizenship to members of the provincial population to gain increased loyalty to Rome and allow some measure of local autonomy by having members of local populations become the emperor¬ís representative in those provinces.