These are rare survivors of the hat trade! 
Fabulous boxes that the hats would have been stored upside down in on the shelving. 
The top of the hat would sit inside the box, as in the photo, so it didn’t get crushed in storage, with the rim sitting on top of the box, hence the fabric around the rim to protect the hat. 
Measurements: 20 x 23 x 15.5cm high