If you buy a hat at Saks,they put your purchase in a hat box.
When I had the hat I wore to my daughter's wedding made (I used an atelier,so that's different),they packed it in a hat box,as they did with her wreath and veil.
But,the last hat I bought,got stuffed into a shopping bag. It wasn't a "good" hat and I bout it when I was on vacation,the store was NOT a department store,so I just figured that's why they used a shopping bag.
That's where my beautiful hat at Nordstroms was destined until I decided to wear it.