If you can get a bank to lend you that amount with a lower interest rate, you can certainly use those funds to pay off the student loan. That said, I wouldn't do it.
Federal student loans have a number of benefits - a variety of payment plans, the ability to defer the loans in certain situations, and the ability to rehabilitate the loan if you default. A loan from a bank has none of those protections, and will probably also have a shorter term, giving you fewer options if you can't make your payments for a few months or even longer.
Your auto loan has a lower rate because it is a secured loan, the bank can take the car back if you don't pay, so there is a lower risk they will be left with the entire unpaid balance if you default. A loan to pay off your student loans would be unsecured, and you'll pay for that higher risk in interest. Unless you have a great credit score and solid relationship with a bank or credit union, I would be surprised to see a lower interest rate than what you have currently.
