The answer is "Yes."
No-fault, if filed timely, will allow you to have your medical expenses paid by your own car insurance company. Your insurance company will determine, from their own investigation, who was at fault for the car accident. If they determine that the driver of the other car was responsible, they will seek to be re-paid by the other driver's insurance company for the medical bills and expenses that they paid on your behalf.
In New York, you only have 30 days from the date of the accident within which to file a claim. If you do not file this claim within 30 days, you will have to have your health insurance company pay for your medical expenses. If you ultimately sue the driver of the other car, and recover money for your medical expenses, your health insurance company may be able to recoup the money that they paid, on your behalf for your medical expenses. This is known as a lien. Depending upon the amount of money you recover for your injuries you, or your attorney, may be able to negotiate and reduce the amount of money that must be repaid to your health insurance company.
In New York an adult (anyone over the age of 18) has three years from the date of the accident within which to start a lawsuit seeking money to compensate them for their injuries. Failure to file a no-fault claim simply means that your medical expenses will not be covered by your car insurance.
If you are on the losing side of the equation, where you're involved in an accident, make every attempt to file a no-fault claim within 30 days of the accident. Once you do that, you'll have your medicals covered up to a maximum of $50,000.