I took the National Geographic test. Still waiting on results, which I hope will provide haplogroup information. I have a theory that I am Norman on the Y-chromosome, or at least a Norman matronymic (independent of Y chromosome, but presumably Normans married Normans first few generations in Calabria). Haplogroup I is clear-cut Norman. However, if I am J2, then this would suggest Greek ancestry from Magna Graecia. Other haplogroups more difficult to assess.

Referring to post way upthread, the Italian who Ancestry.com identified as Jewish, the current prevailing view is that most Askenazi Jews are descendants of Southern Italian women. That is, after destruction of Temple circa 60 AD, many Jews were enslaved or iotherwise dispersed, with male Jews going to Rome and Southern Italy, where they married indigenous Italian women converted to Judaism (yes, way back in time, Jews converted goyim to Judaism).

This Italo-Jewish population was later driven north out of Italy, above the Alps, to the Rhineland, and then driven later still to the East, to Poland, Ukraine, etc

But after the initial phase of inter-breeding with Italian women, the Askenazi Jews became highly endogamous and did not mix with outsiders. Today most Ashkenazi are at most 4th or 5th cousins of one another.

And because 2000 years is a very short time-frame genetically speaking, DNA tests can mistake Italians with no Jewish ancestry for actual Jews because the Ashkenazi are descended from Italian women. In terms of genetics, the two groups are very close