A little history:
From the beginning of the Roman Empire in 27 BC until about 285 AD, it had been ruled by one Emperor. There were times when there was a Co-Emperor, but usually only when the senior Co-Emperor was much older. In 284 AD Diocletian came to power. Some say he was the son of a freedman, a former slave. He rose through the ranks to become a cavalry commander under Carus. When Carus and his son Numerian died in Persia, he was proclaimed Emperor. However, Carus’ other son Carnius was also proclaimed Emperor. After their armies battled each other Diocletian won.
It is thought that Diocletian looked at recent history and saw that a sole ruler did not help the stability of the Empire, as the competing claimants fought against each other and not the enemies of Rome. At this time, around 285 AD Diocletian named Maximian as Caesar or Junior Co-Emperor, while Diocletian retained the title Augustus or Senior Co-Emperor. A year later Maximian is elevated to Augustus with Diocletian and is given command of the Western half of the Empire while Diocletian takes command of the Eastern half. Later two Junior Co-Emperors are named, one for each Emperor. It is thought that this is the pre-cursor to the Eastern Roman Empire and the Western Roman Empire.
My Timeline:
In my time line, everything is good until Diocletian takes power. At this point he still sees the problem of only one ruler and the stability of the Empire, however, he sees that this could lead to the fracturing of the Empire, causing it to fall faster. Divide the Empire, divide both your military strength and your tax income. This is where the timeline split. If the Empire did not split into two, then there was enough money coming in from taxes, and enough Legions to help defend the Empire.