Interesting. My first instinct was to tell you the two events weren't contemporanious, but it turns out they were.
The Huns were active players from about 130 to 453 CE. They enslaved the nearest german tribes (most notably the eastern Goths, aka: Ostragoths), and pushed a lot of the rest into Roman territory. Their own armies at one point (451) made it as far West as Orleans.
To say that they were a major factor in the German invasions of the Roman empire is if anything understating things. While adventurous German tribes like the Ostragoths could expand eastward, there wasn't a big problem between the two peoples. However, in about 370 the Huns rose up, enslaved the eastern German tribes (including the Ostragoths), and all the remaining German tribes ended up huddled together in a little strip of central Europe between the two empires. They clearly couldn't stay there, and frankly attacking Rome was much less forbidding than attacking the Huns (and their German clients).
The Anglo-Saxon migrations started happening around the 400's. Considering the Jutes and Angles completely disappeared from their homelands (replaced by Danes), it might be reasonable to assume they were pushed by something or someone.
Based on all this, I could see (and in some cases have heard) the following possible arguments:
- Advent of The Age of Cavalry. Cavalry had developed to a point where it was so superior to infantry that the matchup between the two was hopeless. A settled farming society had no hope raising a horse-riding army that could compete against pasoralists who naturally spent their whole life in the saddle. The problem here is that I don't think the Germans who invaded England were predominantly pasoralists. However, the Huns were masters at it, and perhaps the domino fall they set in motion ended with coastal farming German peoples skedaddling to a safer side of the English Channel.
- Disease. A series of plagues ravaged the settled areas of the Roman Empire, leaving them relatively depopulated and easier pickings for the Huns and Germans.
- Climate change/Environmental Degredation. something caused population declines in Western Europe at this time. If it wasn't disease, perhaps it was climate change, or perhaps they'd exhausted the land, either of which would cause their old farming techniques to not support the population they used to. Whatever it was, something caused 5th Century Europe to have about half the Romans, and way more Germans, than the 3rd.
- Economic reasons. There are a whole lot of various economic decline theories floating around. Some of them are interesting, but IMHO a country that loses half of its population is naturally going to have economic issues. Better to look at where all the people went.
There are a lot more theories floating around, but these seem the best supported.