Would I win an accident case when both parties have no insurance?

The correct question is whether you'll ever collect from uninsured parties. Insurance has no effect upon winning but does affect collection.

Sometimes the question isn't whether you can win, but whether it's worth pursuing. Yes, if the other car pulled in front of you are hen most likely you will prevail in court. But with no insurance on either car it is doubtful that this would be worth the time,expense and effort to do so.

There may be no insurance policy to collect your damages. However, the other driver may have another source of $$ for you to go after if you win. If you have substantial injuries, it's worth having a lawyer do an asset check.

Insurance has no bearing on liability, but have a local personal injury lawyer investigate.

The simple answer is yes, you may win. However, there are many other factors you should take into consideration before making your next move. Having a court award you damages is not even half the battle when dealing with an uninsured motorist. You then have to actually collect the award from the defendant. This can prove to be very difficult, especially if the uninsured motorist doesn't have any assets. You will need to weigh the costs of pursing the action against your chances of succeeding in the suit, and most importantly of actually collection any award. Any experienced car accident attorney will be able to run an asset check on the uninsured motorist which will allow you to make an informative decision regarding how you should proceed.