Yes, they were called "agnomens"

"The agnomen (plural agnomina) formed an additional part of some Roman names, usually following the cognomen. Usually they were nicknames acquired at some point during the lifetime, but, rarely, some agnomina were inherited."

Examples:

Publius Cornelius Scipio Africanus

Publius is the praenomen

Cornelius is the nomen

Scipio is the cognomen

Africanus was the agnomen, given because of his victories in Africa

(Full name) Marcus Aurelius Marci f. Quinti n. tribu Galeria Antoninus Pius

Marcus Aurelius Antoninus is the praenomen, nomen, cognomen

Marci f. Quinti n. is the filiation. It stands for Marci filius Quinti nepos, meaning "son of Marcius, grandson of Quintius"

tribu Galeria indicates membership in the tribe of the Galeria

Pius was the agnomen

"Biggus Dickus" (Monty Python's Life of Brian) https://www.youtube.com/watch?v=2K8_jgiNqUc