You could, but it would be best to give your current auto insurance company a written notice that you are canceling and switching to another auto insurance company. Without a written notice, your current auto insurance company will keep your policy in effect and continue to bill you. If you didn’t pay the bill, you would be running the risk of damaging your credit, which in the long run can adversely affect your auto insurance rates. It is easy to cancel most auto insurance policies by giving them written notice.