You can definitely trade in a car that isn't running, provided the prospective dealership wants it. In fact, some dealers periodically have "push, pull or drag" sales where they invite customers to bring in run-down vehicles as a way to induce business. However, trading in a non-working car at the dealership isn't necessarily your best financial move.
Additional Information
Dealers may offer $500 to $1,000 of trade-in value for a hopeless car, just to help sell you a new one. In other cases, they try to get a good deal on a non-working car that they can fix up and sell used on the lot. A January 2012 CarsDirect article points out, though, that dealers generally won't pay you the value you'd get somewhere else for your vehicle. You could sell it privately as a fixer-upper to a hobbyist that repairs clunkers. You could also sell it as salvage and take the cash in to buy your vehicle.
