They can probably make it seem that way, but you'll end up paying on the new car or your lease.
The negotiation is with the bank, not the dealer. If you trade, just remember that the bank is going to get what was promised to them. For example, lets say your residual was 10k and you have 5 payments of 200 dollars left on your lease. If you trade it tomorrow, your payoff is 11000 bucks.
Also, most banks inflate their residuals at the time of purchase. What I've been seeing lately is most cars are worth less than their residual values on the open market. Do some research before you pull the trigger on your car. You may be able to buy the same car for significantly less.
Best of luck to you.