Questions like these are always impossible to answer precisely, or in any quantifiable sense. We haven't any surviving, fully intact mail shirts from the 6th century, nor can we reconstruct it exactly - the metallurgy, if nothing else, will always be a bit off. There have been some limited tests performed using reconstructed mail, most famously by Alan Williams. But, we can speak generally about what mail did for the warrior who wore it.

As far as we know, the mail being worn in 6th century England would be very similar to that worn by the late Romans. This would consist of a short-sleeved shirt, hip or perhaps mid-thigh in length, made from 10,000-20,000 iron rings, half riveted and the other half solid. While later medievals would wear some kind of padded garment under their mail, there is no evidence for such at this time. Absence of evidence is not evidence of absence; we simply can't know.

As mentioned above, Alan Williams conducted a series of tests using reconstructed mail and weapons. Williams found that one-handed swords, whether used to slash or to stab, were wholly incapable of piercing his reproduction mail sample; they simply can't generate enough force. He found that bodkin arrows could defeat the mail itself, but struggled to penetrate the padding worn beneath it; only if the arrow was delivered with a force of 120 joules was the armor completely defeated. Lance thrusts and halberd slashes required more than 200 joules to defeat the armor. Again, though, we cannot assume that these tests entirely reflect the reality of medieval armor. I should note that he also tested a 15th-century mail sample, which produced slightly inferior results, but I'm hesitant to make any assumptions from that; most things don't work quite as well 550 years down the line.

So, if experimental archaeology can't fully answer our question, what are we to do? Like all those trained in history, I'm going to suggest that we - say it with me - Read The Sources! I'm sad to say that basically nothing survives from 6th century England, so we will have to extrapolate from other sources describing the use of mail. Rather than dig through every primary source I can find looking for references to mail - sorry, I don't love you that much - I will turn to this article by Dan Howard, who has helpfully collated a number of passages referencing the effectiveness of mail armor.

From Ammianus Marcellinus, describing the Siege of Amida in 359:

The Persian infantry found it hard to avoid the arrows shot from the walls by the artillery, and took open order and since almost no kind of dart failed to find its mark, even the mail-clad horsemen were checked and gave ground.

From Joinville:

...and whilst the Turks were fleeing before him, they (who shoot as well backwards as forwards) would cover him with darts. When he had driven them out of the village, he would pick out the darts that were sticking all over him; and put on his coat-of-arms again... Then, turning round, and seeing that the Turks had come in at the other end of the street, he would charge them again, sword in hand, and drive them out. And this he did about three times in the manner I have described.

From Baha'al-Din, writing about the Third Crusade:

...drawn up in front of the cavalry, stood firm as a wall, and every foot-soldier wore a vest of thick felt and a coat of mail so dense and strong that our arrows made no impression on them... I saw some with from one to ten arrows sticking in them, and still advancing at their ordinary pace without leaving the ranks.

From Galbert of Bruges, discussing a 12th century archer named Benkin:

And when he [Benkin] was aiming at the besiegers, his drawing on the bow was identified by everyone because he would either cause grave injury to the unarmed or put to flight those who were armed, whom his shots stupefied and stunned, even if they did not wound.

From Usamah ibn Munquidh, describing an encounter in which he struck a Frankish knight in the side with his lance:

He bent sideways so much that his head reached his stirrup, his shield and lance fell off his hand, and his helmet off his head...he then resumed his position, erect in the saddle. Having had linked mail under his tunic, my lance did not wound him.

Same author:

A Kurd named Mayyah smote a Frankish knight with a lance, which made a piece of the link in his coat of mail penetrate into his abdomen, and killed him.

Also same author:

...the horseman who struck Philip the knight, for verily the Franks have all been astounded on account of that blow which pierced two layers of links [back and front] in the knight's coat of mail and did not kill him.

What can we make of these sources, and why did I assault you with them? Basically, what we can see is that, at least on these occasions, mail could provide prodigious protection from a variety of weapons, and this supports the results of Williams' tests. It was consistently difficult to pierce by archers at long range, and could stand up to lance thrusts under certain circumstances; under other circumstances, it could fail spectacularly. The accounts don't describe its effectiveness against swords, but given that arrows and lances outperformed swords significantly in Williams' tests, I think we can assume that he is largely correct in thinking that swords would have grave difficulty in penetrating mail.

Finally, a note on the purpose of wearing armor. The idea is not to make one's self invulnerable; that is patently impossible. Humans, given enough time, will always find ways to get through the armor. The idea is not to give them the time. Armored men simply have a much larger margin for error than unarmored men and can take risks that the latter can't. I've used the metaphor of a car's seat belt before. Even if you wear your seat belt, there are some collisions that simply aren't survivable. But, by wearing your seat belt, you greatly lower the odds of that happening. An accident that would have killed you becomes one that seriously injures you; an accident that might have seriously injured you becomes one that you walk away from with bruises.

It's the same principle with armor. A battle is not a duel. Whether in a small-unit skirmish or the rarer pitched battle, in which you are fighting shoulder-to-shoulder in a close formation, against enemies in a similar formation, most blows will not be perfectly aimed and delivered because the fighters' main priority is not dying. It doesn't matter how bad ass your mail shirt is if someone cuts your hand off or stabs you in the testicles; you're out of the fight. But, where even a glancing blow will injure a wholly unarmored man, an armored man can keep on trucking. This makes you a much more effective fighter because you have the self-confidence to fight more aggressively than someone who isn't armored.

Works Cited

The Knight and the Blast Furnace, by Alan Williams.

Mail: Unchained, by Dan Howard.