They have lids under their lids

Usage examples of hatboxes.

For a start, the amount of luggage was sufficient for a stay of several months, and comprised a number of old-fashioned and very bulky hatboxes, an awkwardly shaped leather case which Pretty clung to as though her very life depended on doing so, a large trunk which required two men to lift it, and a variety of suitcases.

She went to Harrods quickly then, and bought more dresses in less time than she had ever done in her life, got her hair done nearby, and took a cab back to the hotel, laden with hatboxes and dresses, and her new hairdo.

Nathaniel moved a stack of hatboxes and she made a place for herself near the hearth.