Plenty, especially if you look at the so called barracks emperors of the third, fourth, and fifth centuries.

• Pertinax is the first, being the son of a freed slave.
• Maximinus Thrax was seen as a foreign barbarian of low birth, even if claims that he was a Goth are largely discounted.
• Pupienus was the son of a blacksmith.
• Aemilian was of low North African heritage.
• Claudius II "Gothicus" was from an Illyrian family of no note.
• Aurelian was the son of a peasant farmer.
• Diocletian was the son of a freed slave. Actually there are versions where he himself was a freed slave.
• Maximian was the son of a shopkeeper.
• Galerius was the son of a Thracian herdsman.
• Severus II was the son of an Illyrian farmer.
• Licinius was the son of a Dacian farmer.
• Vetranio was the son of a Moesian farmer.

There are dozens of others who we don't have good family backgrounds on. I've also skipped a lot of emperors who's only claim to nobility was being related to one of these commoner emperors. Plenty of these also are from dubious sources. Aemilian's family is vaguely mentioned at best. And a number of these come from decisively antagonistic sources.

Regardless, in the third century at least, power came from the point of the sword. Any man of skill with an army at his back could aspire to the title of Augustus, no matter how lowly his birth.

After the first 2 dynasties (Julio-claudian,Flavian) there were men of low birth that did become emperors.Pertinax (reign January 1, 193 AD – March 28, 193 AD),Aurelian (reign September 270 AD – September 275 AD).Especially during the crisis of the third century many claimants to the throne were of lower birth (usually military commanders).Also Diocletian, one of the most noteworthy roman emperors, was of low status at birth. It wasnt uncommon for lower class romans to rise through the military ranks and claim the throne.

There certainly were especially as the empire began to degrade. Following the assassination of Severus Alexander and the end of the Severan Dynasty, a number of emperors known as the barracks emperors took over. Many of these emperors were of the commoner class who had risen to command armies exemplified by Maximinus Thrax, the first barracks emperor. He had joined the military as a common soldier however he developed a reputation for himself through his military prowess. Maximinus was disliked by the Senatorial classes as evidenced by the multiple attempts on his life however he suppressed several of these attempts to overthrow him by powerful senatorial families, he was eventually killed by his own troops while sieging the city of Rome.

The period known as the Crisis of the Third Century was riddled with such emperors of common or of middling social status at birth. In total, 14 of the 22 emperors that reigned during the 49 years of the Crisis were barracks emperors and while not all were low born, they tended to be from the class of commoners who joined the military as soldiers.

The Third century was one of tumult for the Roman Empire as it began its march into decline. During the 49 years of the Crisis in particular, those who commanded a powerful army could take control of the Empire and many did, causing massive social and economic upheaval the the concurrent civil wars and lack of stable governance took its toll on the institutions of the Roman government.