My great grandfather immigrated from Italy, but because my grandfather was born before my great grandfather became a US citizen, under Italian law my grandfather is an Italian citizen, my mother is and Italian citizen and I am an Italian citizen. If I fill out a bunch of forms and gather a bunch of documentation, I can get an Italian passport. That doesn't mean I'm applying for Italian citizenship. Under their law, I AM an Italian citizen. So I was born here, my parents were born here, and my grandparents were all born here. Am I a Natural Born Citizen, or does Italy--a country no one in my lineage has lived in for 96 years--control my US citizenship status?
By all considered theories of NBC I am aware of, you are a an NBC. But you grandfather would not have been under the most authoritative interpretation as I understand it. Other considered theories would have your grandfather as a NBC, assuming you meant he was born in the US of A.
NBC: “natural born citizen”.
If your parents were citizens, then yes you are.
Your grandfather wasn't however, since his father was not a citizen at the time of his birth. He was a citizen though, via the 14th amendment. I suspect my great grandfather Fred, actually Hiram Fredrick, was in the same situation as your grandfather. Born in the US, but his parents not yet citizens. Some of his siblings were themselves naturalized citizens, having been born in Swabia (now Germany), but he was native born, he may have had younger siblings who were natural born.
The real criteria is not whether you can claim or be claimed by some other country, but wether your parents were citizens at the time of your birth, and where you born here. (With some exceptions for children of diplomats and military).
Barack H. Obama's father was not a citizen at the time of his birth, or ever for that matter.