In fact, referring to the books, it appears that Boromir did not even have a mail-shirt:
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The Company took little gear of war, for their hope was in secrecy not in battle. Aragorn had Anduril but no other weapon, and he went forth clad only in rusty green and brown, as a ranger of the wilderness. Boromir had a long sword, in fashion like Anduril but of less lineage, and he bore also a shield and his war-horn.
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Gimli the dwarf alone wore openly a short shirt of steel-rings, for dwarves make light of burdens
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But then I often find when I'm watching the films that I see incidents where a man ought to have died but did not, or vice versa, in battle. Surely the fight arrangers are just as pedantic as anyone in knowing about such things, so it does surprise me a little.