There is no legal requirement that a defendant or insurance company must settle a case, even to prevent it from going to court trial. If your car in not drive-able without being repaired, point out to them that owe you for the reasonable cost of a rental vehicle until your's is fixed, and it is not limited to the daily rate and time limit they have to pay their own insurance. Point out that the police report finds their insured entirely at fault and it is clear the driver of the car in front of you has absolutely no fault [they were not backing up, or i assume coming to an unnecessary, sudden stop], so there is absolutely no need to speak to the other driver, who probably does not know anything about the accident beyond they were rearended. If they will not immediately start the settlement process [and your damages do not exceed $10,000], you will just file suit in Small Claims Court which will upset their insured.
Answered on May 19th, 2017 at 8:31 AM