Can your jaw bone end up sticking out of your gums after having a wisdom tooth pulled...?

No, not normally: When your wisdom tooth is pulled, there should not be any part of your jaw bone sticking out. Best to see the oral surgeon to confirm.

Yes: Sometimes, depending on complexity of surgical extraction, pieces of jawbone may separate (break away) from the jaw, and it is very difficalt sometimes to detect due to bleeding during the surgery, so after surgery those pieces of bone, called bone sequesrum, show up and sometimes it is very painfull or causes lots of discomfort. It may cause also bad smell, or bad breath.

Sequestered Bone: Yes. At times after the removal of a tooth, as the bone heals and remodels to a new shape, small peices of bone can brake off the rim of the thin alveolar socket where the tooth was and travel to the surface of the gum. They may fall out on their own but it is best you go back to the dentist or oral surgeon who did the surgery and have them removed. It is a common and benign condition.

Bone fragments: The answer is yes. When teeth are removed, sometimes the bone from which they were removed either needs to be cut or drilled, and sometimes even when that's not necessary, small fragments may come loose and "exfoliate" even days to weeks later. In most cases, a simple exam and conservative care is all that is necessary.