Italian citizenship can be passed down through ancestors--kind of like genes that are inherited. Before 1948 only men could pass it on, not women, and it cannot be passed on once the person becomes a citizen of another country if he renounces his old citizenship at that time (as naturalized citizens of the US were required to do before about 1970). So in my case, my great-great grandfather came here from Italy in around 1890. My great-grandfather was born in 1895 in NYC, and the Italian citizenship was passed on to him from his father. My great-grandfather then became a US citizen in 1898. Then my grandfather was born in NYC in 1923, and my father in 1952, then me.
The only obstacle I have encountered is that while Italy as we know it was united in 1861 (thus the rule that ancestors born before 1861 who emigrated could not pass on Italian citizenship), evidently the "Republic of Italy" was not really formed until after 1898, when my great-great grandfather renounced his citizenship. The Boston Consulate, who in my case is the final authority on our eligibility, told me I would have to make an appointment and bring in my documents from Italy to have a certain answer, mostly because it goes so far back.
The guy I spoke to at Boston was not very helpful really; when I told him it was my great-great grandfather he first said "that's too far back" but when I explained in more detail and mentioned what the New York consulate said, he said if I could prove the ancestry, it would probably be okay but that I needed to make an in-person appointment and bring the papers in.