The bandbox had nothing to do with a company of musicians, and is a common misconception.
But, back in the seventeenth century, it was a box of thin wood, chip, it was called, in which lords and ladies kept the very wide collars or “bands” of lace or plain linen that were then in vogue.
When starched to stand up, they were simply “bands,” but when extending flat across the shoulders and down the chest, they were called “falling bands.”
Our Puritan forebears are depicted with small plain falling bands, and the clerical collar of today owes its origin to that band.