Their rights are in your loan documents. You can pretty much guarantee those terms provide that they can refuse partial payments and demand payment as they want. You need to get a written agreement resolving the amounts due, or they will shortly take your car and the lender or a collection agency will then sue you for any deficit.
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I agree with the previous answer and once you are in default, ( missed a payment date) they do not have to work with you according to most loan documents.
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Your contract with the finance company will control, but generally speaking they can probably refuse under-payments where you've already missed payments and have identified you're having financial trouble. It is arguably a commercially reasonable practice to demand assurances from a borrower she can keep current on a large loan (such as car loan.)
As to whether refusing your money is a good-faith business practice, that's another story.
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Acceptance of payments after breach may be construed as a waiver of the prior breach and prevent them from proceeding with the repossession. Of course, as other attorneys have stressed, the loan documents control and should be reviewed by an attorney to determine your rights in this case.
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