There can clearly be liability there depending on the facts and circumstances. There can also be comparative fault between both the bicyclist and the driver, again, depending on the facts. The case is definitely worth being investigated.
Hopefully there is a police report that memorializes the event but generally speaking, you do have a potential claim worth pursuing further. You should contact a qualified personal injury attorney in your area as soon as possible.
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With a C-2 fracture, you really need good representation. That is a serious injury and you should find someone in your area who does a lot of personal injury work and retain someone quickly. good luck.
Personal injury cases only; I'm good at it; you be the Judge! All information provided is for informational and educational purposes only. No attorney client relationship has been formed or should be inferred. Please speak with a local and qualified attorney. I truly wish you and those close to you all the best. Jeff www.nyelderinjurylaw.com
"No contact" cases can be tricky at trial (and almost always lead to a denial of claim by the no fault carrier). This description, however, looks like a very makeable case. Was the cyclist wearing a helmet? Was there any head injury along with the c-spine injury? How far from the car was he when he jammed on the brakes? When did he first notice the car? What does the driver state on the accident report? There are dozens more questions, but this is a start. Find local counsel, maybe in Rochester.
The opinions expressed in this answer are not intended to be taken as legal advice. These opinions are based on New York practice. I may be contacted at 212-553-9300.
You may very well have recourse and a good case. Call a personal injury attorney at once.
This answer is not intended as legal advice, nor as a substitute for legal advice received from an attorney during a consultation. Your rights and obligations may vary depending upon facts not disclosed in your question. Therefore, you should contact an attorney immediately so that your interests are properly protected.