Yes, but this course of action rarely proves to be fruitful. The sentiment is understandable; you were left injured by someone driving illegally without insurance. However, one must consider the conditions for which an individual drives without insurance. Typically, people drive without coverage due to financial difficulties. Thus, the odds of being able to collect any suitable damages from an uninsured driver is usually low.
This is why having adequate uninsured motorist coverage on your car insurance policy is so important. Despite the fact that you might not be able to recover damages against the other driver, your insurance may be able to provide compensation for your damages. The increased premiums for thousands of dollars of additional coverage can sometimes amount to as little as an extra $5-10 per month.