It is not uncommon for a car to be repaired while the claim is still being settled. When this happens, the insurance company may not pay the repair bill immediately, and that in turn could ramifications for your car. Since your car can be held as collateral until the payment is received, this situation could result in your being left without transportation.
Another option, if your car is being held pending payment on a claim, is pay for the repairs yourself and let the insurance company reimburse you. Call your insurance company first and work out some details on how and when the company will settle, and then pay the repair center directly out of pocket. This will get your car back into your possession, but it will place the burden of inconvenience solely on you.
You may also want to try to negotiate with the repair center. It may be that you can reach an agreement with the company, especially if they do work for your insurance company regularly, allowing you to drive the vehicle while the claim is being settled. Keep in mind that this would be at the discretion of where the work was performed, and they are not obligated to do so. In fact, if the bill for the car is not paid after a period of time, the repair center even has the right to sell your car to pay for the work performed on it.
When researching a car insurance policy, pay attention to how quickly claims are settled, because too much time waiting on a settlement could end up costing you in more ways than one. Even with the company that is fastest to settle, you will run into situations where things are held up, but it helps to know the company is working on your problems as soon as they happen.