I managed to get a bit stuck in my rotary tool. I happen to have a Black and Decker RTX, which is fully compatible with Dremel bits, attachments and the like (and has more torque, or did when I bought it many moons ago). But, I managed to tighten a bit into the chuck too tight to be able to get it out with my wrench with the chuck/spindle lock.
The procedure is fairly straightforward and offered insight as to why the lock slipped with too much pressure.
Step 1: Remove Anything on Both Sides
Make sure and unplug your tool before beginning.
Remove any parts that are attached to both sides of the rotary tool. In this case, the thread cover (the round grey ring pictured) and the hanging ring/ loop used to hang the tool for carving with a flex shaft connect the two halves together.
Step 2: Remove Fasteners
Remove any fasteners that hold the body halves together. These required a security torx. If you don't have these they are cheap at Harbor Freight or they can be ordered online rather inexpensively. I used to think only the phone company and specialty tool shops had these, but they are easy to find. Very carefully pry the body apart. If you feel unusual resistance to the normal dirt and grime holding things together, go back and make sure that you have found all of the fasteners. Sometimes manufacturers put them under stickers, but that shouldn't be the case here.
Step 3: Remove Drive Shaft From Motor
The entire shaft should be one assembly, with the shaft, bearing and chuck and the stuck bit. Lift it up and out gently away from the motor.
You can see that the chuck lock is made from a plastic gear and a plastic switch. With enough force, the tooth on the lock that engages the gear can be forced out of the teeth and force the tooth into the next groove of the gear, turning the shaft and not loosening the chuck. Thankfully I realized that if something inside was popping loose it was better to not repeat said attempts at the risk of breaking something unnecessarily. You will be much better off taking your tool apart and risking it (at very low risk) than to risk breaking off part of the lock inside and having to break into your tool anyway but with broken pieces inside.
Step 4: Remove Bit From Chuck
With two pair of pliers (I recommend one of them being locking pliers to simplify things), hold the shaft with one and loosen the chuck with the other. Presto chango! The bit should be free at last.
Reassemble in the opposite order and you should be up and running!
Step 5: Bonus Tip!
For some of Dremel's tool attachments (like their jigsaw attachment for the Dremel), they require you to buy a different model. Upon inspecting the models in the store I noticed that the only difference in the heads is that the new model had two cutouts in the threads that allow the bosses in the collar to fit and not spin. Creating these is easiest if you have or have access to another rotary tool, but you could also cut them with a utility knife, file, or other means.
The best way to make sure they are the right size is if you have purchased one of the accessories requiring the grooves (the best time to make the modification anyway). They are simply opposite each other symmetrically. Start slowly and only remove as much material as you need so as to not weaken the threads unnecessarily.