The Empire did break up into three states in 258 (Crisis of the Third Century). Aurelian reunited them in 270-274.
Tough it was more a result of situation, rather than a separatist policy in the West.
The so-called Gallic Empire was nothing but the demesne of usurpers that failed to make it to Rome, but that Roman emperors didn't have enough forces to deal with. As soon the crisis over, it was crushed.
You need both will of separation from the Empire (and that's going to be hard, given it' was the main policial and social model) and some sort of legitimacy to do it. Probably as a slow separation that contemporaries wouldn't be that aware of, but pointed out historiographically (as ERE).
A good candidate would still be Palmyre, but differently than OTL. The Palymerian rulers were trusted with such titles as "Dux Oriens" (Eastern leader) : Odenadathus was basically the hegemon of the oriental provinces safe on northern Turkey and Egypt.
With a more important Barbarian pressure, an harsher IIIrd crisis, you could see some weird form of Tetrarchy appearing, with every dux/caesar having more autonomy than its counterparts, the emperor being more or less ceremonial in these matters.
For example, in the west, maybe
Bagaudae could evolve, with a weakened imperial power (or defeated one, that wouldn't as IOTL adapt to little war issues), from provincial rebellions, into autonomous group of provinces acknowledging the imperial power but having its own.