Depending on your definition of 'someone', I would say that it was relatively easier for Romans to rise in society. Romans who possessed full citizenship could rise politically and socially through their own merit. Unlike today, doctors and actors were not quite 'high on the totem pole', so to speak. Instead, Romans could serve as soldiers and rise from there. Prestige and social status were deeply intertwined with military rank.

As such, an excellent example of a Roman rising above his birth is Belisarius. He was born to poor Thracian parents, about as low as you can get. As a youth, he became a soldier and Emperor Justin I's bodyguard, then the leader of the Emperor's bodyguard regiment. After the death of Justin I, he was given command of an army, and eventually, through his achievements, was chosen to command the Roman expedition to retake parts of Africa. This was one of the greatest military excursions made by the Romans during this period. For his victories during this campaign, he gained the last ever Roman Triumph, and is known today as one of the 'Last of the Romans', a paragon of Roman values.

Another example is Trajan. Though he was not born into poverty, he was not Roman. Instead, he was born in what is now Spain. Like Belisarius after him, he joined the army as a way to advance. After serving honourably, he became Consul at a young age. This is partially attributed to his father's position as a patrician. Still, it is supposed based on his military merits that he was named successor as emperor by Nerva, the reigning emperor. After Nerva's death, Trajan became emperor; one that was so popular he was bestowed the title of optimus (meaning 'the best') by his peers.

Finally, a slightly iffy example would be Cincinnatus. He also was not born into poverty, but due to legal problems he was forced to sell his land and essentially live in poverty, working a farm with his own hands. When a Consul died, Cincinnatus was elected suffect consul. Due to how ably he performed in this position, it was Cincinnatus that the senate turned to when the Republic was in peril. He was granted the position of Dictator, which gave him absolute power. Though he willingly gave this position up after defeating Rome's enemies, for a time he was the most powerful man in the Republic.

Sources:

The History of the Decline and Fall of the Roman Empire

The Cambridge Medieval History - Volume I: The Christian Empire