This is a frequent Avvo.com question. What typically happens is someone without bodily injury insurance hits someone who doesn't have uninsured motorist coverage. The person who was hit visits several attorneys, and they all tell her that they can't represent her because there's no way for the attorney to get a recovery. Maybe you're in that position right now. If you are, well, I'm sorry. I cannot help you either.
Law firms are businesses, too, just like any other business. Personal injury attorneys like me make our money off insurance companies, through either bodily injury or uninsured motorist coverage. If neither exists, then we won't touch your case. It's a bad investment.
So, can you sue the other party anyway?
You can sue, but our courts award judgements, not money, and it's up to you to collect on those judgements. Most people who carry only basic insurance are "judgement-proof." In other words, they have nothing, so you get nothing. Litigation isn't free, either. You'll need to pay the filing fee, the process server fees, expert witness fees, and a myriad of other costs. In the end, you might have a worthless judgement that you will never have any hope of collecting on unless the defendant wins the lottery. I wouldn't hold my breath hoping for that to occur.
Here's my advice: Take that money and buy insurance, and buy as much stacked uninsured motorist coverage as you can afford. You'll be glad you did it the next time some other uninsured driver hurts you.