Thank you so much for your answer. By the way, as I said, our case was that it was originally "Chernetski", "Chernetzky", "Zernetzky", etc.. Still, we changed it to "Czerniecki" when we Polonized it, and then changed the "e" to an "a" when we tried to pose as Poles. Also, as Dr. Dara Horn notes, immigrants generated their own name identifications:
True, European Jewish immigrants did have to render their names into Latin or Cyrillic letters..., and yes, passports were sometimes forged-but those...changes would have been generated by the immigrants themselves. It is also true that many immigrants chose new names for themselves..., whether for expediency or to avoid discrimination.
PS Why do both "Czarny" and "Czerny" exist for "Black" in Polish?