Strange timing. My mom just got into a car accident the other day. Very different...she was driving down a regular road and a guy blew a side road stop sign and t-boned her. My advice is the same.
yes, you can sue...more than likely you can even get some sort of compensation for lost time, wages, pain and suffering etc. That is really the norm anymore.
Now my honest opinion? Are you hurt? Is your car NOT going to be fixed because the insurance company has denied your claim? Has this injury impacted your life in a way that debilitates you to an extent that you are no longer able to properly function at your job or future jobs?
If yes, to any of those sue. At least this gives you legal documentation for any further medical claims that could come from this.
If not, no, I personally do not think you should. Why? Because the suing for every single accident has a direct correlation for the constant increase in the cost of insurance. The point of insurance is to protect you and your assets from damage that you would normally not be able to pay for. If they are doing this, you have been taken care of and there is no reason to sue.