You can, but you need to be able to prove that they failed to maintain the vehicle, that they leased a defective vehicle or that they negligently entrusted the vehicle to your sister. As much as you may not like the thought of doing so, you would also need to bring a claim against your sister as it is likely that any defense would include that the accident was the fault of the operator, and not anything wrong with the car itself. There could also be a potential products liability case; however, that would be extremely costly to undertake.
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You would have to show that the rental car company failed to properly maintain the car. You would also have to show that you suffered "serious injury" as that term is defined in the NY Insurance Law. Be sure to fill out a no-fault insurance form within 30 days of the accident to get your medical bills and lost wages paid.
I am a former federal and State prosecutor and now handle criminal defense and personal injury/civil rights cases. Feel free to check out my web site and contact me at (212) 577-9797 or via email at Eric@RothsteinLawNY.com. I was named to the Super Lawyers list as one of the top attorneys in New York for 2012. No more than 5 percent of the lawyers in the state are selected by Super Lawyers. The above answer is for informational purposes only and not meant as legal advice.
This sounds like a horrifying incident. I am glad to hear that you walked away relatively unscathed. The answer to your question is: Absolutely yes.
A rental company has an obligation to keep it's vehicles in good repair to avoid exactly these types of situations. From what you describe, they have completely failed in their responsibility to maintain the vehicle rented by you and your sister. Specifically they failed to periodically maintain and check to see that the brakes on this vehicle were safe.
In no way is this a "slam dunk " type case but there is a claim worth looking into based upon the injuries that you and your sister have suffered. The rental company will likely try to pass the blame on to the vehicle manufacturer or some other party.
This is something that you probably do not wish to hear, but if you decide to pursue a case you may have to sue your sister. I am sure that this was not your intention at all but the truth is that a jury may find that her actions as driver of the vehicle in some way contributed to the happening of the accident. If you both brought a claim you would likely need two different lawyers as it may be construed a conflict for the same lawyer to represent you both.
Regardless it sounds as though you have something worth pursuing and I suggested that you immediately contact a local attorney to help guide you through the process. Do not give any statements to the insurance company for the rental agency as they will try to use this against you later on in the litigation.
Also you are both entitled to no-fault benefits but given that the accident happened out-of-state some investigation would need to be performed to see who is responsible, whether the rental insurance company or the carrier for your own vehicle.
The aforementioned opinion does not constitute legal advice and is for general educational purposes only. See an attorney licensed in your jurisdiction for competent legal advice. No attorney-client relationship has been formed through the within legal question and answer session.