Some good responses on here and I agree. I'll add that when you trade your car in at the dealer, the dealer already has a very good idea of what they'll get for that car (or if they'll get anything for your car), and they are already negotiating that into the price you'll be paying for your new car as you're sitting there trying to get money off. They're ahead of you.
Once in a while you may run into a dealer that is somewhat level with you. Or at least you they may be halfway level - my last buying experience felt like that - they told me my trade in would go to the auction and fetch between $5,000-$7,000 grand...that they would try to sell it to an employee first. I felt the same way. I figured the most I could get for the car in a private sale was $8,000. They gave me $6,000 for the trade...they initially offered me $4,500.
In the end I'm sure I lost, but I was buying a new truck with a lot of rebates. They said I got a "great deal". The manager told me that, personally (lol)...said they didn't make much off of me, but that they were a high volume dealer and they would sell ten of these trucks this week...that it was a rainy day, the owner was on the property and personally had to approve the deal because...it was slow for a Saturday...yada yada yada. I actually kind of believed him.