Hire an engineer - or see if the insurance carrier will hire an engineer - to see if the foundation is okay.
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.
I agree with Mr. Rothstein. Hire an engineer to confirm that there is no damage. If there is damage, and your insurer will not cover it, then you can bring suit against the neighbor.
If you'd like to discuss, please feel free to call. Jeff Gold Gold, Benes, LLP 1854 Bellmore Ave Bellmore, NY 11710 Telephone -516.512.6333 Email - Jgold@goldbenes.com