Rump states! What are they? Where did they come from? Where did they go? Cotton-eye Joe?
In short, a rump state is a leftover bit of a bigger country or kingdom that still identifies itself as part of that polity, and so do its neighbors. The boring examples of this are the UK being a rump state of the British Empire. Very exciting, ooh ahh. So I'm gonna ignore those in favor of what I think is cool: when Empires fall, are there any provincial bits left over that still retain their Imperial government for a time?
It should not surprise you (or it may, I don't know you), that yes, that kind of things happens. Now, I'm not talking about when a neighboring power comes in, conquers a barony and makes it their own with the same name. That's your bog-standard collapsing kingdom procedure. I'm talking a local governor is able to maintain control while their parent goverment goes to lunch and doesn't come back.
By the time the fifth century rolled around, the Western Roman Empire was foundering. Territory was being lost left and right to Germanic tribes, some of whom had even taken effective control of the Empire as the powers behind the throne. Emperor Majorian inherited a fractured Empire where areas of now Switzerland, southern France, and southern Germany were under the control of various Germanic tribes. He was able to recapture large amounts of territory, reuniting the province of Gaul with the Imerpial center. However, upon his death after four years of rule, much of that reconquest disintegrated. This left Gallia geographically isolated from its parent Empire.
Roman Gallia was governed by the magister militum Aegidius, appointed as such by Majorian in 458. After the death of Majorian, Aegidius continued ruling the province and was able to defend it from barbarian attack. Upon his death in 464 or 465 his son Syagrius succeeded him, taking the title dux, traditionally given to a provincial military commander. Despite this, the neighboring barbarian kingdoms referred to him as King of the Romans.
In 476, the Western Roman Empire finally fell when Orodoacer conquered Roman Italy and established his own kingdom. Syagrius did not acknowledge Orodoacer's kingdom as legitimate and severed all ties with the Italian peninsula. Thus, the former province of Gallia was now on its own.
Syagrius was able to rebuff barbarian invasions for ten more years, and the whole time maintaining that he was merely the local authority of a Roman province. In 481, Clovis I ascended to rule the Kingdom of the Salian Franks, the eastern neighbors of Gallia. In 486, Clovis defeated Syagrius at the Battle of Soissons and took all his territory. And thus, the last Roman territory, ruled by Romans in the Roman way, was gone.