A full Roman name, to identify an individual (and appropriate for one’s tombstone) had a lot more than three parts. It could include:

First Name if male
Gendered form of family name
son/daughter of Father’s-full-name
Father’s title if illustrious enough
son/daughter of Mother’s-name if impressive enough
Husband’s name, optional
Tribe name, optional
Personal Name if female
Nickname or Honorifics if male
Names from Adoption

…it could go on and on.

…and then there were rules for freedmen and women to show who their former masters were and their ties of obligation.