The amount of money that a dealer (bank) will lend on a car loan depends mainly on your credit, income, debt service ratio etc...
The amount of negative equity you can add to a new loan depends on your credit, and on the new vehicle you're buying, as well as the rebates on it.
For example, if your credit is decent and you qualify for a new car, a bank might lend up to 140% of MSRP on a new vehicle. Meaning iuf you're buying a $20k car, you can get a loan for $28000, leaving you some room to add your negative equity.
However if that $20000 car has a $4000 rebate, then you'd be still financing the $28000, but you'd be able to "bury" $12000 in negative equity, because the car would really cost $16000.
The bad thing is you'd be owing $28000 on a $16000 car, in which you'd be buried basically until the last day you pay it off (60, 72, 84, maybe even in 96 months?). It all depends on the loan, and the terms. That's why as others have pointed out it's not really worth it.
In the short run you might be paying less monthly, and drive a newer car, but in the long run you'll end up paying more cause you're paying interest for your other 2 cars.
So will a dealer do it? If you get a good approval, and there is a vehicle out there with enough rebates to cover your negative equity, then yeah, it's doable. Not all manufacturers or dealers will do it as they each have their own finance rules, requirements etc... so you'll have to call around. The question is is it worth it for you to actually do it?
And keep in mind unless you both have perfect credit or a ton of money down, you might as well forget about "your" picks such as hot sellers, because chances are they won't fit the bill. You might only be able to to the swap on slow sellers with huge rebates.