Can a finance company threaten to repossess car?

Yes, depending on the state in which the vehicle is domiciled and the loan is originated.

In general, if after thirty (30) days a payment is not made on a vehicle then a finance company may notify the borrower that they are going to reposess the car.

For customers with good credit, the threats do not come until after sixty (60) or ninety (90) days. For customers with historically poor credit, the threats will come as soon as possible (30 days).