Uliss is not allowed sticks anymore for this very reason, she seems to have the sort of shaped mouth or bite that lodges sticks up in the roof of her mouth very easily.
The first time when she was a puppy, I couldn't understand why she wasn't eating and the vet couldn't either. He looked in her mouth and down her throat and completely missed the stick (hopeless vet in France.) She didn't eat for days, I was spoon feeding her yoghurt and roasting her chickens, anything to slip down which didn't touch the sides. Eventually I took her outside in the really bright sun and shone a torch in her mouth and discovered the stick, why the vet missed it I don't know. Anyway she was on anti inflammatries for a couple of weeks and eventually I got her eating again. She had lots so much weight and had become afraid to eat.
It seems whenever she snaps a stick in bits she gets some stuck up in the roof of her mouth as it has happened loads of times since and now I always check her mouth and take sticks away from her. We never had this problem with previous cocker and he was passionate about his sticks on walks.