In addition to Mike's answer - in *The Hobbit*, the coat Thorin chooses for Bilbo is described as

 >"...wrought for some young elf-prince long ago. It was of silver-steel which the elves call mithril"

For it to fit Bilbo, it had to be a very young Elf prince. The idea that there was only ever one mithril chain mail shirt forged and it was given to a young elf, who would soon outgrow it, seems very unlikely.