The answer is yes, but it will crack.
When you roll and fold a Panama there is always some breakage of the palm fibres it is made from. If your hat is warm, moist and supple then it will be minimal, but in the long run your hat will crack and be disfigured.
If you have a Montechriti fino like below, in the traditional centre crease Optimo shape, then your chances are greatly improved. However most people dont like the look of the centre crease (it is a bit odd), and dont want to spend $500+ on a fancy hat that will end up being broken.
In the old days people did roll them, but only in an emergency, because they will always make your hat look wonky. Usually these hats had no hat band, or just a simple thin ribbon. They also bought them for a lot less than we do today, so just didnt mind when they broke. Often you will see photos of the Optimo crown looking pristine and crisp - thats because they have never been rolled.
Manufacturers make a rollable hat using different techniques :
* The super-bleached rubbery panama. This one feels like plastic and is surprisingly flexible, though it tends to look cheap and rubbery. They will roll for a while and usually crack through the crown.
* Twisted weave open crown Panamas. These will roll reliably for a good time, but they dont look very nice and feel cheap.
* Dishonesty. This is the most common solution, Sell people a hat that will roll for a while, and dont worry if it looks like crap on their head and will break. People come into our store all the time lamenting the tourist Panama that cracked.
It is a big part of the lore of Panama hats that they can be rolled, and while it is actually true, it just isn't a good idea in the long run. It is usually a trick. As a Panama specialist I have been very keen to work this out - it just does not work well enough to be done. I would feel bad selling them.
A better solution is learning how to pack them, or choosing hats that can be crushed. Check out the video below: