Strictly speaking, dub-dub, straw for hats is plaited, although plaiting is so similar to weaving that I wouldn't usually pick that particular nit. (If for no other reason than for fear of trespassing on Faldage's territory!)

I have seen what Might be described as a "straw bowl", Jackie, made out of strips of plaited straw, would round and round and sewn together ( in a similar manner to a coiled pot) Some restaurants over here use them for bread or rolls, offered whilst you are waiting for your meal to be served.