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Okay, then here's my case: My great-grandfather immigrated from Italy 100 years ago. My grandfather was born here before my great grandfather became a US citizen. Under Italian law, Italian citizenship was passed to my grandfather, who never renounced it (since he didn't even know he had it), to my mother, and to me. If I come up with the documentation of that chain--birth records, marriage records, naturalization papers--I can get an Italian passport. So, I'm born in the US. My parents were born in the US. My grandparents were all born in the US. But under Italian law, I'm an Italian citizen. Are you saying that I'm not a natural born US citizen?

Claims on Italian citizenship are only possibly valid for descendents down to the second degree. Your grandfather born in the U.S. could have inherited Italian citizenship from his Italian citizen father, and your mother could have inherited it from your grandfather, but you are past the second degree and are therefore ineligible.

More, here: Italian Governement: Citizenship FAQ

And here: Consulate General of Italy

Even a second degree descendent would have to serve in the Italian armed forces, become an employee of the Italian Government, or live in Italy for at least two years after reaching legal age to be eligible for Italian citizenship.

Here's what the website of the Italian consulate in Miami says:

Who is entitled - categories
If you were born in the United States or a Country other than Italy you can be recognized an Italian citizen if any one of the categories listed below applies to you: 1) your father was an Italian citizen at the time of your birth and you never renounced your Italian citizenship;
2) your mother was an Italian citizen at the time of your birth, you were born after January 1, 1948, and you never renounced your Italian citizenship;
3) your father was born in the United States or a country other than Italy, your paternal grandfather was an Italian citizen at the time of his birth, neither you nor your father ever renounced your the Italian citizenship.
Your mother was born in the United States or a country other than Italy, your maternal grandfather was an Italian citizen at the time of her birth, you were born after January 1, 1948 and neither you nor your mother ever renounced your Italian citizenship;
4) your paternal or maternal grandfather was born in the United States or a country other than Italy, your paternal or maternal great grandfather was an Italian citizen at the time of his birth, neither you nor your father or mother, or your grandfather ever renounced your/their Italian citizenship. Please note: a grandmother born before 1/1/1948 can claim Italian citizenship only from her father and can transfer it only to children born after 1/1/48.