I tried this suggestion out, and definitely felt something move inside. I have owned many iPhones, and knew this wasn't normal. So I decided to take a"look".... Sure enough, the battery was loose from the glue or what ever sticky stuff they use to hold it in place. The ribbon cable that connects the battery to the phone is very thin and fragile, so if there is any battery movement at all, it will fray and break in no time. Im my case, the ribbon was broken, therefore not fully starting. It put the phone into recovery mode automatically. In order to restore the phone, iTunes needs the phone to respond during the restore, but with the ribbon broken, the phone cannot, and an error (2001) pops up. You're stuck, until apple replaces the battery. Don't try doing it yourself, if you have warranty, otherwise it will be voided.
That being said, for all of you that 'shaking the phone worked', you're on the verge of that battery ribbon breaking!