It happened notably against Attila, but then it was too late for the Empire. They probably liked their Chinese leaders a lot more but they were unused to the exercise of arms and after conquest didn't want liberty enough to take lessons in a hard school. Chinese did it, but it took them 1 century.
Systems always work until they fail.
Absolutely, and at the end all systems are doomed to fall, even the best. Or just enhance what could be in current ones. Barbarians were however a bit more than simply a mob, imo. ROme experienced mob in the capital and inr ebellious provicnes and usually beat them up, even when trouble hit the Empire. Some years later Belasarius was able to show just how weak the barbarians really were when even a few of the people were prepared to resist.