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.
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.
Neil Kokemuller has been an active business, finance and education writer and content media website developer since 2007. He has been a college marketing professor since 2004. Kokemuller has additional professional experience in marketing, retail and small business. He holds a Master of Business Administration from Iowa State University.