It is a matter of record that the Saxons invasions of Britain began in the last years of the Roman occupation, and then started in in full force after the Romans left the island in 410 AD.  They marched away, seemingly without a backward glance, leaving the Britons–after 400 years of occupation–to fend for themselves. Map retrieved from: http://historiarex.com/e/en/225-anglo-saxon-invasions

From Gildas, writing in the 6th century:

From Britain envoys set out with their complaints, their clothes (it is said) torn, their heads covered in dust, to beg help from the Romans. … The Romans … informed our country that they could not go on being bothered with such troublesome expeditions; that Roman standards, that great and splendid army, could not be worn out by land and sea for the sake of wandering thieves who had no taste for war. Rather, the Britons should stand alone, get used to arms, fight bravely, and defend with all their powers their land, property, wives, children, and, more importantly, their life and liberty. Their enemies were no stronger than they, unless Britain chose to relax in laziness and torpor; they should not hold out to them for the chaining hands that held no arms, but hands equipped with shields, swords and lances, ready for the kill. This was the Romans’ advice.

http://www.ict.griffith.edu.au/wiseman/DECB/DECBps.html

These invaders, as the map shows, were not in fact all ‘Saxon’, but a combination of Jutes, Angles, Saxons, Franks, and Frisians, each hailing from a different region of the western coast of Europe.

The Angles, Saxons, and Jutes were Germanic peoples.  From Wikipedia:   They were “originally a small tribe living on the North Sea between the Elbe and Eider Rivers in the present Holstein. Their name, derived from their weapon called Seax, a knife, is first mentioned by the Roman author Ptolemy (about 130).

In 3rd and 4th century Germany, great tribal confederations of the Alamanni, Bavarians, Thuringians, Franks, Frisians, and Saxons arose. These took the place of the numerous petty tribes with their popular tribal form of government. With the exceptions of the Saxons all these confederations were ruled by kings; the Saxons were divided into a number of independent bodies under different chiefs, and in time of war these chieftains drew lots. This leader the other chiefs followed until the war ended.

In the third and fourth centuries the Saxons fought their way victoriously towards the west, and their name was given to the great tribal confederation that stretched towards the west exactly to the former boundary of the Roman Empire, consequently almost to the Rhine. Only a small strip of land on the right bank of the Rhine remained to the Frankish tribe. Towards the south the Saxons pushed as far as the Harz Mountains and the Eichsfeld, and in the succeeding centuries absorbed the greater part of Thuringia. In the east their power extended at first as far as the Elbe and Saale Rivers; in the later centuries it certainly extended much farther. All the coast of the German Ocean belonged to the Saxons except that west of the Weser, which the Frisians retained.”  http://en.wikipedia.org/wiki/History_of_Saxony

From then on, there are differing views about how rampant the Saxon spread was.  Certainly, it happened (language alone tells us that), but the exact timeline for the spread is not clear.  The Battle of Mt. Badon (whether or not fought by Arthur) is said to have occurred around 500 AD, which held back the Saxon tide for a generation.

After that, however, it was unstoppable.