Am I able to get my full down payment back

If the dealer was at fault in some way, you should be able to recover your down payment and any other costs you incurred in reliance on this (such as any registration fee[s]). The key is, the dealer must have been at fault in some way, such as by lying to you about the loan status (e.g. telling you that you were approved when you were not). If they were provably at fault and do not voluntarily compensate or reimburse you, you could sue them for money (e.g. in small claims court, "pro se," or as your own attoney) with a reasonable chance of success (no case is ever guaranteed).
But if the dealer were not at fault, they would not have to return any money to you. So, say that the dealer had told you that it looks like you qualify for the loan, but that the loan was not fully approved or underwritten yet (or something like that); in that case they did not lie, and are not responsible for any losses you suffered due to the failure to be approved, since you would have been aware that the loan was not final and therefore there at least some economic risk in going foward. Or say that you provided incorrect or incomplete information initially, and were disapproved when the correct information was determined; or that something changed in the interim, like losing a job, taking out other loans (which affects your debt to income ratio), etc.--again, they are not liable if what they told you was true when they told it.
Sothe issue is, to recover money, you have be able to show that the dealership was at fault, which generally means showing a lie or misrepresntation.