You can have a claim against your own personal UIM carrier and the at fault drivers bodily injury policy. Contact a local AVVO attorney to assure that you are not personally dealing with the insurance carriers.
It depends on what is the nature of the settlement with your insurance. If it was under the uninsured motorist line of coverage, your insurance company would have subrogation rights against the at fault party and you would not be able to sue them. Typically this line of coverage is not invoked unless the other party does not have enough insurance to cover your injuries, however, so you wouldn't typically settle with your own carrier unless the other party's insurance has already paid out or if they don't have bodily injury coverage.
There is the possibility that the settlement with your own company was for a PIP lost wage claim, in which case you would still be able to make a claim through the other party's insurance.
I recommend you consult with an attorney who will be able to investigate. Many will do so free of charge to you.
I am not sure what you mean by "paid settlements". An injured person has potentially 4 different claims in a car crash like what you describe. Claim 1 is with their own PIP insurance, which pays medical bills and wage losses. Claim 2 is against the at fault driver for that person's BI insurance. Claim 3 is against the injured person's own UIM insurance, which may be responsible if there is no insurance for claim 2 or if the claim's value exceeds the coverage in claims 1 and 2.
Normally, the analysis ends there, but your scenario opens up possibility number 4, a claim against the driver of the car they were occupying. If it was a 2-vehicle crash, caused by the neglect of both drivers, and if the injured person was occupying one of the cars, he or she has the potential for up to 4 claims.
So, even though I am not sure exactly what "paid settlements" you are referencing, the answer to your question is "yes". Whether that claim has merit or value will depend on many factors, and should probably be reviewed by a lawyer.
In an automobile accident the at fault driver's policy a.k.a. the BI claim is the first source of ins. money for a settlement. However, each person has the option in their own policy for uninsured/ under-insured motorist coverage. S people may have had their own UM coverage tender or settle as they may have evaluated the personal injuries above what the BI limits were. That's a typical scenario.
The answer to the question depends on a number of factors. One of those factors is the terms of the settlement agreement, to include what parties were included in that agreement. Another factor is the coverages provided by the insurance carriers at issue. While Florida Law, in essence, mandates that all owners of vehicles have No-Fault Insurance (not that everyone abides by the law), it does not require all drivers to carry bodily injury coverage. This means there is no guarantee that an at-fault driver has any insurance to cover the victim(s) in an auto accident. However, policies that carry bodily injury coverage may have uninsured/under-insured motorist coverage. This sometimes helps out victims of accidents where the at-fault driver has no or limited coverage for his or her liability. Also, sometimes, individuals may carry umbrella coverage on other policies that they own, which could provide for additional coverage beyond the auto policies involved.
I believe Mr. Reynolds hit it on the nail with his response, however there is one thing I'd like to point out if I understand your question correctly: once you have settled for the value of the claim and there is no residual claim due to exhaustion of benefits or any other policy term, then you cannot claim again.
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