Very few accurate tests have ever been performed on this subject. Actually, I would even go so far as to say that no accurate test has yet been performed.

The closest test to being an accurate one for 15th century mail was performed by Alan Williams in The Knight and the Blast Furnace. He tested both a 15th century piece of mail and a replica of it made by Erik Schmid (one of the best makers of replica mail), and found that both resisted thrusts from a spear, a blow from a halberd and an arrow quite well, although the original mail and the padding underneath was penetrated fully at 120j, which is within the range of the Mary Rose longbows (and whether or not those are accurate in a 15th century context is not a can of worms I want to open).

With that said, the test was not performed using an arrow, but with a simulated arrowhead and, I believe, a drop machine. The dynamics of an arrow are very different, as the flexing of the shaft can reduce penetration. We also don't know whether the rings were hardened, if they were all riveted, or what kind of steel they were made from, so it's not possible to apply these results backwards in time to earlier mail.

What we can know for certain is that the arrows of the Saracens were not particularly useful against mail. We have references to crossbowmen in the Third Crusade with numerous arrows sticking out of their mail or gambesons who were entirely unaffected, and we have gems like this paragraph from Jean de Joinville:

Once when they flung it at us, it fell close beside the tortoise-tower that my Lord of Courtenay's men were guarding, and buried itself in the river bank. And presently comes a knight, named " the Albigensis," and: "Sir," says he to me, "unless you help us, we are all burnt; for the Saracens have let fly so many of their fire-darts, that it is just like a great hedge all ablaze bearing down on our turret." We jumped up, and hurried to the spot, and found that he had spoken the truth. We put out the fire, and before we had got it under, we were covered from head to foot with the fire-darts that the Saracens shot across the river.

I'm not well read enough to know of a similar source referring to the ability or inability of Western arrows or crossbow bolts to penetrate mail, but it probably depended a great deal on the range, the angle, the type and quality of the mail and the energy of the arrow. The same would apply to thrusts from sword or spear. Obviously it must have done some good against them or else it would have been abandoned and, likewise, it can't have been perfect or there would not have been a need to reinforce it first with leather, then replace it with plate armour.

Another commented has already done an excellent job answering the question. They did mention that few or no accurate tests have been conducted, which I agree with. To elaborate though, here are a few common issues with tests so you can make a more informed judgement next time you see a video about it:

• Quality of items: Most modern costume mail is butted, which is to say the rings are bent into circles and left that way. This is not how historical mail was made, as it is very easy to bend them back open again with even a moderate thrust. Historically links were riveted shut, or made of alternating solid and riveted rings. It helps to know the type of steel used in the reproduction, and whether that was comparable to mail of the time period being examined. Likewise for the style of weave (4 in 1, for example), the thickness of the steel and the diameter of the rings. Very rarely do I see tests that provide this information at all, so it's hard to know whether the mail tested really resembles anything like what was worn in period. Same goes for the weapon tested. Is it representative of the period in shape and weight? Was the steel hardened? Should it be? In the case of the longbow, the bodkin is often viewed as the anti-armor point of the day, but since most surviving examples are made of unhardened steel or iron, this view is being called into question.
• Is the mail being put in context? Testing mail by placing on a block and hitting it is not representative of mail worn by a person. Blocks of wood don't absorb forces the same way a human body does. Even if it did, mail was essentially never worn by itself. Any tests on mail should be performed on top of a good representation of a gambeson or other garment we know went under it, and the material under that should be a good representation of skin, fat and muscle.
• Battlefields are not static. Look at the way the blow was struck. Was it a big, wind up thrust straight at the chest? How likely is it a soldier would be able to pull off a strike like that against an active target? That represents an absolute ideal strike, as most blows are likely to be quick hits at oblique angles. A target may stagger under a heavy blow, absorbing force much more than if they were to brace themselves against it, like a static target against the ground or leaning on a tree. Perhaps that represents a coup-de-grace against a downed foe. But it's certainly not representative of typical fighting.

Finally, let's forget the science and think in terms of common sense. Essentially every person who was able to afford mail used it. There was virtually no debate that its protection was worth the added weight. Its widespread use over a huge area and a great period of time is proof of its effectiveness. On the other hand, if it was utterly impenetrable then there would be no point in the later development and use of plate armor.