One factor that can lead to the breakdown of the materials used to make a straw sun hat is improper storage. If you crumple it, pile other items on top of if it, or cram it in a drawer, it could very well lose its shape and structure. The best way to protect a sun hat made of straw, straw-like material, or braided paper blends is to hang it on a hook or rack or store it in a hatbox.