Initially, I cannot tell if you are asking for yourself or your mother. My answer will refer to you being the driver of one the vehices. The analysis is basically the same whether it was you or your mother. I assume from your "Question" and the "Question Detail" that by your statement "neither insurance company wants to accept responsibilities for the damages to the damage" you mean the damages to the vehicle they do not insure (i.e. You are insured by Acme Insurance; Acme Insurance does not want to pay anything to the other driver involved; and the same holds true for the other driver's insurance company to you). Basically, each insurance company believes that the driver they do not insure is at fault (or they believe fault is 50/50). Assuming you had "Collision Coverage" with your insurance company, your insurance company would pay to repair your vehicle minus your deductible. Your Question does not refer to whether anyone was injured. If you were injured, your insurance company would pay your medical expenses if you have "Medical Payments Coverage." They would pay your medical expenses up to your limit under this coverage. If the other driver's insurance company refuses to pay you for the damage to your vehicle and any other damages (i.e. rental vehicle, medical bills, lost wages, pain and suffering, etc.) then your only option is to sue the other driver. Assuming the insurance company does not change its mind after your lawsuit is filed, a jury ultimately would decide who is at fault or the percentage of fault by each driver. I mentioned fault being "50/50" above. In Nebraska, to recover against another driver you must be less than 50% at fault for the accident. If fault is 50/50, you do not receive anything from the other driver. If the jury comes back with your fault being less than 50%, then you receive what the jury decides your case is worth less your percentage of fault (i.e. Jury decides case is worth $10,000 and you are 25% at fault; you receive $7,500). If your own insurance company refuses to pay you, assuming you have the appropriate coverage as discussed above, then you would have a breach of contract claim against your insurance company. The way I read your question, this does not seem to be what you are asking though.
Answered on Jan 10th, 2012 at 11:08 AM