Can you file a claim on both insurance companies yours and the party at fault in an auto accident?

Yes, you can. However, ultimately it is the At Fault party's insurance company that will pay for the claim. In such a case, you will file a claim under your collision coverage and be subjected to the deductible. Once the claim is settled, your company will subrogate the loss with the other company(i.e. get reimbursed). Once that happens, you will be provided your deductible back. If you go through the at fault party's policy, you would file the claim under the property damage of the policy.