I find stems like this (and even larger) in maybe one in 30 - 50 sticks. If I get a really bad draw, and cutting/massaging doesn't help, I go looking for a big old stem. Dig a little in the head with my tiny needle nose pliers, and many times, if you are careful, you can pull it out without damage. The stick usually opens up beautifully. Sometimes though, if the stem is really big (1/8"+), you have to squeeze the head to a wedge shape to reduce the draw. Just had to do this a few minutes ago with a Graycliff 1666. No draw at all, and I ended up removing a freak'n twig. After that, great tasting smoke. Sure beats tossing them.