There are two important versions of this story. The first one was written by Gildas, probably in the early 6th century (the dating of his work is very uncertain), in a very curious work called De Excidio Britanniæ (About the Ruin of Britain) ; the second one is incorporated in the Historia Brittonum, which was composed in the 9th century. The Historia Brittonum provides the standard story of the evil Vortigern; however, it was written well after the events and is hypothesised to have been a part of a slander campaign against a Welsh dynasty claiming descent from him.
Gildas' story about the adventus saxonum is certainly “streamlined” (and actually does not mention Vortigern, though Bede later provides the name; it is now hypothesised that older versions of Gildas might have included it) (i.e. it is not an exact description of the situation; for instance, some Germanic warriors/leaders probably served in Britain as a part of what was remaining of the Roman army there), but the basic situation it describes does sound plausible in itself. Actually, a great specialist of post-Roman Europe, Walter Goffart, called the take-over of the W. Roman Empire by barbarians “a policy experiment gone wrong.” Calling German mercenaries to protect the population against Pictish and Irish raids, and attacks from other Germanic peoples, was at that time perfectly sensible.
(sorry for the brevity of this answer, I hope someone will be able to elaborate; otherwise, I might supplement it a bit later)
During late antiquity and the decline of Roman state power pagan tribes of germanic and of other cultural-ethnic origins were hired or settled throughout the whole of the Roman provinces. It's noteworthy that these peoples had served in the Roman army since the time of Augustus. Also migrations and attempts of settlements had occured since 105 bce (Marius) at the latest. So these "barbarians" were known to the population and the magistrates of the provinces. Referring to this, it is interessting that some scholars think, that Vortigern is a mutated version of Gwrtheyrn = celtic for overlord, which might have denoted some kind of britanno-celtic magistrate.
For Bede the pagan Anglo-Saxons are the godly scourge, who chastices the degenerate Britons for their sins. This said, it is obvious why he stresses the accounts of warfare, intrigue and violence commited on both sides. So, all of his writing is to be taken cum grano salis, although he's an historian of good repute from an overall perspective.
Relating to the settlements of the Anglo-Saxons in GB it is also noteworthy that it is quite difficult to draw the line between Anglo-Saxon and Britonic and/or Welsh/Celtic territory, custom and culture. There's also vivid discussion within the scientific comunity, whether and to what degree the different cultures and ethnicities mingled and merged and, for example in the case of burial sites etc., what burial objects constitute an allocation of the buried person within one of the mentioned cultural-ethnic spheres. So, it is quite difficult to solve the question, whether the Anglo-Saxons (and the always omitted Jüts :-) ) were cultural-ethnicaly constituted peoples before crossing the channel in one or two huge migrationial waves, after Vortigern's invitation, or if a huge number of heterogenous small bands of people migrated to GB during some centuries, settled and worked as mercenaries and with time evolved into significant peoples or ethnicities after undergoing some kind of ethnogenesis. Following the second assumption, this people, which constutited itself after the migrational period, would have constructed the story of the three distinct tribes/peoples crossing the channel in retrospect, because they were oblivious to such scientific concepts as ethnogenesis and it was en vogue to bolster up one's own legitimacy with made-up genealogies and stories.
Further reading: I found the chapters "The Social and Political Background" (P. 1) and "The Anglo-Saxon world view" (P. 66) in "The Cambridge Companion to Old English Literature" very compelling.