I answered a question a while ago about the differences between freedmen and citizens in ancient Rome. There was also another post that asked about the differences between American slavery and older forms that you may find useful.

I hope that is a helpful start, but I'd love to go more depth about your question specifically. In my senior thesis on Greek and Roman slavery, I likened the slavery of antiquity to a mire of grey as compared to the more recent black and white dichotomy of the antebellum South. Ancient slaves, citizens, and freedmen held endlessly shifting positions of power, reputation, honor, and financial success. Aristotle, wrote a wonderfully succinct passage on the topic, he claimed a common phrase of the times was:

“One slave comes before another, and one master comes before another.” (Aristotle, Politics 22)

Just as citizens had a social hierarchy, so did slaves. Slaves according to their job, talents, abilities, etc. had the possibility to find opportunity and advancement. A slave well versed in navigation and shipbuilding, although much more expensive and rare than a common laborer, stood a much higher chance of amassing a fortune for his master and earning his gratitude. This gratitude could manifest itself in many forms, but small freedoms and liberties were common rewards that could eventually lead to manumission in the future. In fact, if you read my previous posts the imperial bureaucracy and power structure was abound with slaves and freed slaves who held tremendous power and wealth (often called the domus or familia caesaris).

Most scholars, one of them being K.R. Bradley, are careful to note that even though some slaves may have had some opportunities the majority did not. Bradley said as much when he wrote:

“But the fact remains that in spite of Rome’s liberal practices most of the servile population probably never achieved freedom at all.” (Slaves and Masters in the Roman Empire, p.83)

So the Romans had much lenient practices concerning their slaves than other societies at the time, but still achieving freedom in your life time was pretty unlikely. A slave who had no skills and worked as a laborer on just one of many of the plantations that his master owned stood little to no chance of social benefit. However, slaves who were active parts in the master's household or had more valuable skills that caused them to interact with the family more often (i.e. fabrication of goods, tutoring ability, financial ability) would have increased odds of both social advancement and eventual freedom. However, Bradley also warned that we should try not to impose our more modern notions of universal emancipation on antiquity:

“The neat cleavage between masters and slaves, between those who were free and those who were not, should not lead to any belief that slaves in the Roman world ever formed a rigid social class, in the modern sense of that term, with a recognizable consciousness of itself and a concomitant programme for its own amelioration.” (Slaves and Masters in the Roman Empire, p.15)

The ancient world had no standard of a true slaveless society and the slave "class" was so inherently diverse and fractured that even if it could imagine such a world, it would have no effective method of working together to arrive at such a vision. Slaves, as a general rule of thumb, were not working together to advance themselves, but often undercutting one another just as free rivals do.

Perhaps the most important thing to take away from all this is that even though Rome's manumission and social advancement was no where near ideal, it was considerably better than their counterparts. Even more so, despite their failings, some Romans seemed to be well aware of how gracious they were as conquerors, as seen in the following quote from Emporer Claudius:

"What else proved fatal to Lacedaemon and Athens, in spite of their power in arms, but their policy of holding the conquered aloof as alien-born? But the sagacity of our founder Romulus was such that several times he fought and naturalized a people in the course of the same day! Strangers have been kings over us: the conferment of magistracies on the sons of freedman is not the novelty which it is commonly and mistakenly thought to be, but a frequent practice of the old commonwealth." (Tacitus, Annals 11.23-24)

The Romans who, at least according to Livy's Early History of Rome, came from extremely humble origins (i.e. a collection of rabble, debtors, fugitive slaves, etc.) often liked to view themselves as a very fluid society for all peoples and, as Claudius argued in that passage, some believed it was what made the Romans greater than other civilizations. The reality often fell short of this lofty ideal, however, we must always consider how important this ideal was to the Roman psyche.

To summarize, Roman slaves did possess the ability to climb up the socio-economic level, however, the social fluidity was completely contingent upon the will and circumstances of the master and the abilities/skills/value of the slaves themselves.