I'm sort of one of them, too. My mother converted to Orthodox Judaism in the 1940's, from an Italian background. She then met and married my dad (a born Jew), and I was born about 15 years later.
In the 1980's when I got into genealogy, I found out from a distant cousin on her side that their family has Jewish origins via Spain (they had fled to Sicily and Italy after the Inquisition in Spain). Lots of clues, things I never picked up on but should have (such as that my grandmother would burn two candles on Friday night, calling it "an old family custom"; or that my grandmother refused to eat pork or have crucifixes in her home...she said pork was "dirty and full of worms" and that crucifixes were "bad luck".)
I never picked up on all those clues which I now know were converso clues. I just thought they were Italian things (Grandmom also used to tie a red string to things when they were new, like a house doorknob or a car...I later found out that is from the Kabbalah...and Sephardic Jews were/are heavily into Kabbalah.)