This ca 1840-1860s style bonnet is either a small child’s bonnet or a large doll bonnet. If the latter, it may date from later.
Arriving nearly flat, likely how it was stored, it will take some gradual coaxing to get it closer to its original shape. I already started adjusting the wired neck edge, which feels like it only has one break.
The brim is drawn on narrow cane’s. I estimate they are at most 2mm thick. These appear to all be intact. There doesn’t appear to be any decent on the sides of the brim, nor shaping for cheek tabs. But, with the amount of compression and possible up-shifting of the cane’s, this shaping may not be in place. Hopefully, when more depth is restored, the shape will be clearer.
The interior shows a stiffened, loose weave cotton in the crown as well as an organza like material with a shimmer to what I think is dried, aged stiffening agent. The silk crown lining was originally gathered into the crown. A band at the top of the brim has bows of narrow ribbon.
The bavolet is deep. It needs to relax before an accurate measure can be taken. It is set on the grain rather than on the bias as was common in the mid-nineteenth century. The bavolet is lined with a very thin silk organza. There is a fold line parallel to the lower edge evident in one area that may or may not be a former hem fold.
Conservation plans: Slowly restore depth with support with archival tissue. Hopefully, I have enough archival foam left from straw bonnets to make a small support for this bonnet. Relay the shifted layers. Re-gather the crown lining. Re-tack the loose section of the band inside the brim.
If I Had My Own Blue Box: 
Oh, lovely! I wonder if it is a doll bonnet?