Geez, I haven't financed a car for 8 years, but the one time that I did not pay cash I made certain the financing was approved PRIOR to handing over my trade-in and driving the car off the lot. That is the best thing to do if you can.
However, I saw this happen to a friend of mine and this is how it went. He bought a car and drove it off the lot and left his trade-in at the dealership. Three days later, they called and said the financing fell through and that he'd have to bring the car back. So he did expecting to be given his trade-in and downpayment back. Uh. . .no.
They had already shipped his trade-in off to a wholesaler. They offered him a car from their used lot that wasn't even close to what he had wanted to buy stating that, that was the kind of car they could finance for him. They flat out refused to locate his trade-in and give it back.
They had the nerve to start the financing paperwork for him on the used jalopy and had applied his downpayment for the new car to the used one and thus could not "release" the down payment.
If that wasn't bad enough it got worse. He demanded an explanation and a copy of his loan application. The salesman had neglected to include his shift differential pay to his annual income which made a significant difference (this is why you should always double check you application and not let the salesman "help" you with it). He pointed out the error and the salesman told him he was still out of luck because he signed it and refused to resubmit it.
As luck would have it, the owner of the dealership was in for an annual spot check and my friend grabbed him and told him what was going on. After much discussion and threat of a lawsuit, he got to keep the new car after verification was made that there was more income than stated on the application.
I drove my friend home to get his W-2 and he presented it. Not good enough, verbal verification was still needed. 2 days later they told him to pick up the new car and that financing had been approved, but they stuck him with a hefty application fee (he did sign the incorrect document and was out of a car until he agreed to pay it, so he did) and he had been driving a rental car.
If there had not been an error though, to answer your question, yes the dealership took the car back.