From the article I quoted in my above post:
"An ingenious correspondent has suggested that, as wooden sticks are known as billets, the stick-biting practice might have first been called 'biting the billet' which then modified to 'biting the bullet'. Worth investigating certainly, but my initial enthusiasm was dampened rather by finding that no records of the phrase 'bite the billet' can be found in historical newspapers and other records. That would have been a neat solution but, lacking any supporting evidence, it seems best to discount it."