Depends on the type of claim, if it was a Bodily Injury claim and you signed a release you can't because you relased that person with the settlement you signed.
In most instances, an insurer will require you to execute a "release" before any insurance proceeds are paid. This document will prevent you from seeking any further relief.
You need to find out what the statue of limitations is in your state. This means that you have a certain amount of time to sue. It is different for bodily injury then it is for medical claims.
Depending on the circumstances of the claim itself and how the claim was handled by the parties involved, it may be possible to file suit for a higher claim settlement. I would advise hiring a public adjuster to go through the claim to see if anything was left out of the original settlement before filing suit.
When a settlement is reached, the agreement usually includes the signing of a waiver stating that no further claims can be made concerning this matter. If that agreement has been signed, then the person would have immunity from further suits.
You can sue for costs paid that were not covered by your insurance. You can sue for pain and suffering or lost wages, if this was not covered by their insurance.