It's a touchy issue, but if they do, it won't be for at least a few more weeks, because they will do it orderly. The goal of bankruptcy is to get some help to get back on their feet stronger, so they will make sure that they can do it right.
The government won't let them completely die though, so even if the GM name disappears, which it probably won't since that will destroy every bit of recognition they ever had, GM will still be around in some way, shape, or form.