Being upside down on your trade in is very, very common. The term upside down means the same exact thing as negative equity. This means that you owe more on your trade in than it is worth (or so the dealer says). So, how do you trade in a car with negative equity?
With people financing vehicles for longer terms to get lower payments, some as long as 7 years, it's very common to have negative equity. This is also true due to dealers putting people in cars that are not worth near what they are being sold for. Dealers call that "knocking someone's head off". The way that this is done by defrauding the lender by describing the vehicle on the loan application as having an upper end trim line, bigger engine or more features than the vehicle really has. This adds "loan value" to the car and allows the dealer to sell it to you at a higher price. You end up oweing much more for the vehicle and in some cases, thousands of dollars more than it should have ever been sold for in the first place.