The tetrarchy was an attempt to provide each part of the Empire with a ruler and to establish an ordered, non-hereditary succession. It broke down when his dominating personality was removed (OCD 346).The tetrarchy was inherently unstable and when Diocletian left the scene, civil war immediately broke out.
Oddly enough, through the entire period of the tetrarchy Rome was neglected. The imperial residence might be in Nicomedia, or Treviri, or Sirmium, but Rome, the origin of the empire, was a neglected afterthought. To be sure, the tetrarchy gave lip service to Rome, but not much more. But this was nothing new as Rome had been neglected for years, through most of the third century in fact. H. H. Scullard notes that "while the armies played the game of emperor-making, the security and unity of the Empire was nearly destroyed" (OCD, 932). The economy largely did and so did politics. So much for Rome as an ideal. The fact that Aurelian (270-275) had to build a wall around Rome showed that while the emperor might reign from the oceans to the stars (Virgil, Aeneid I.287), there were plenty to molest and make afraid. Scullard notes "increasing difficulties in communication, and above all the general sense of insecurity and lack of confidence" (OCD, 933).
Nevertheless, the tetrarchy had a convenient scapegoat, and they could always blame their personal failures on the Christians and use it as an excuse to persecute them. Christians ever after would remember this time as the great persecution. Their fondness for Constantine, who later replaced Constantius in the tetrarchy saved the tetrarchy from bearing the appropriate brunt of the ill will of the people. Diocletian alone would be remembered for it, even if it was his right-hand man, Galerian, who was probably more behind the persecutions.