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 out grow it seems very unlikely.