So I get that your first name was your given name [Gaius, Lucius, Marcus, Marcus,] I get that your second name was your clan [(Gaius of the clan) Marius, (Lucius of the clan) Cornelius, (Marcus of the clan) Antonius, (Marcus of the clan) Tullius.)
But then what was the third name as in (Lucius Cornelius) Sulla, (Marcus Tullius) Cicero, or (Gaius Julius) Caesar? It can't have been a patrician/plebeian thing because Marcus Antonius and Gaius Octavianus were both from patrician families, whereas from what I understand Marcus Tullius Cicero was from a plebeian or equiiites family.
I know sometimes it was an earned nickname like Pompey Magnus or Pompeius Strabo. But if it was a nickname, why would it be passed down certain families (Gaius Julius Caesar, father to Lucius Julius Caesar and Gaius Julius Caesar,) but not others (Gnaeus Pompeyus Strabo, father to Gnaeus Pompeyus Magnus?)
The third name element in a Roman name from Republican or early imperial times is the cognomen, originally a nickname - like Caesar, Scipio, Bestia, Cicero, Crassus and so on. These served often as further identifiers for a certain branch of a gens, since they tended to become hereditary, so for the Caesar branch of the gens Iulia, the Iulian clan or the Nero branch of the gens Claudia.
These names are sometimes very old, but never a necessary part of the official nomenclature (the praenomen, the name of the family and the name of the father were a necessity) but their importance increases only in the late Republic to reachs its peak in the early Empire, when practically everyone had a cognomen. Earlier, the individual name + family name was enough to differentiate individuals, especially considering the fact that Rome for a long time was basically a big city with lots of territory. The cognomen gradually gained importance, first mainly among patrician families, while several plebeian families still lack one even in the late Republic, such as the Antonians (Marcus Antonius) or the Marians (Gaius Marius), so it kind of could be seen as a patrician/plebejan thing (Marcus Antonius and Gaius Octavius were both plebejans - there is a patrician branch of the gens Antonia, which is where the confusion might come from).
So while three names were increasingly commonplace, and came to be known as a characteristic of Roman citizens (see also How did Roman names work?, they were not a hard rule.
As a side note, if you refer to Augustus as Octavianus, you are using his third name, his cognomen, which was the name he gained upon the adoption by G. Iulius Caesar. From then on, he was Gaius Iulius (Divi filius, son of the divine) Caesar Octavianus. Since an adoption entailed the legal transfer to another family, the name would change as well, and one would lose one's original family name, but would usually add a cognomen modelled after the original family name; such as Publius Cornelius Scipio Aemilianus, who had originally been a member of the Aemilian clan. Augustus himself probably never used that name, but others did when referring to him, such as Cicero.