I've definitely seen programs on the telly with people shooting through leather, mail and plate armour from considerable distances with longbows.
Most of the mail around nowadays is massively inferior in quality to the mail that would be found on a medieval battlefield. Wikipedia has this to say about modern mail usage:
Mail is now used in protective clothing for butchers (against meat-packing equipment), scuba divers (against shark teeth) and animal control officers (against animal teeth). The British police use mail gloves for dealing with knife-armed aggressors. The military also uses mail vests for the same reason. Modern re-enactors of medieval battles and living history also use mail in combat.
In the little combat use that mail armor sees nowadays, it is not required to deflect anything more dangerous than knives or animal teeth. In re-enactment, the re-enactors are either using fake weapons not designed to kill, or following a strict coreography in order to avoid maiming each other. Good-quality mail requires more time and effort to make than poor-quality mail, and understandably, most people and organizations are not willing to pay for good-quality mail when they can get a cheaper product that is adequate for their purproses. Good-quality mail is thus mostly made for wealthier re-enactors who want to pay extra to use something as close to the real armor as possible. The demand there is limited, thus little good-quality mail is produced. As a consequence, it can be hard to get hold of, and because of this, and often lacking expertise on the subject, the people doing the tests usually end up using poor-quality armor. This should be true for plate armour, as well. I will note that at least one History Channel program that included tests like this has used butted mail, which is greatly inferior in every aspect compared to good-quality riveted mail that would have been used by knights throughout the medieval period.

In addition, usually mail was worn over padding. The padding might be easily penetrated by an arrow when worn alone, and an arrow might penetrate mail significantly, but tests indicate that when they are worn together, the arrow imparts a sufficiently large part of its energy to the mail that the padding may offer enought resistance to stop it from penetrating significantly. This fits in with the contemporary accounts of numerous arrows sticking from mail-clad soldiers. Again, good-quality padding consisting of numerous layers of cloth is probably rather difficult to find in the modern world.