The biggest new car financing mistake buyers make is trying to buy a new car without researching online auto loan rates, or knowing if their credit history can support a car loan. Buyers see "0% APR" ads, but can't get the cheap auto loan rates because of their low credit score. Applying for new car financing without knowing your credit score is the dumbest thing a new car buyer can do.
Occasionally people who traded in a car they owed money, were surprised 2 months later that the new car dealer did not pay off their old car loan in 10 days as promised. You are still responsible for making payments on your old car, and effectively will be forced into making two car payments a month. If you trade a used car in for new car loans, make them put in writing they'll pay off your car loan in 10 days, or no deal.
If your current car is worth less than the new vehicle, dealers arranging financing might take your old (current) balance and add it to the new car price to combine them into one loan, which can make your monthly payments higher than you expected.
Always car loan payment calculators (there are many you can use online) to verify car loan interest rates and payments.