The answer to your question depends partially on what time period you are considering. During the Roman Republic many of the generals were of high birth. These include famous historical figures such as Julius Caesar, Scipio Africanus, and Pompey the Great. All these men were indeed of senatorial rank, the highest echelon of Republican society, though there were exceptions to the rule: Gaius Marius, from whom the Marian Reforms take their name, was of equestrian birth - while by no means "poor", he was of a lower social class than the senatorial elite.

When considering the Roman Empire the question of timing is equally important. The Roman Empire was founded in 27BC and lasted in the East until 1453 - nearly 1500 years. Throughout this period, there were a significant number of men who ascended to the highest echelons of society from the lowest. For example, during the 200's AD there began a series of Emperors known to history as the "Illyrian Emperors". These Emperors all hailed from in and around the roman province of Illyria, in the Balkans. These Illyrians were largely poor provincials of humble birth who joined the Roman military, eventually working their way up the military hierarchy to become generals before ascending to the rank of Emperor.

From the time of the Illyrian emperors towards the end of the Roman Empire in the West, the army was largely made up of poor provincials and later, barbarians from beyond the empire's borders. Senior roles were thus often filled by men of humble birth: the emperors Aurelian, Maximinius Thrax, and Justin I are some examples of poor provincials rising to the highest levels of the Roman military and government. To get to the heart of your question, is is difficult to say precisely what the chances were of humbly-born soldier rising to the level of a Roman general. However, the facts indicate that it was certainly possible, at least at certain points in Roman history.

What can be said is that the army was a gateway to power, with many military men wielding immense power in the Roman empire. These include not only provincial Romans who ascended to the purple, but also barbarians from beyond the empire's frontiers such as the generals Ricimer and Aspar who effectively ruled the Western and Eastern halves of the Roman empire in the 400's AD through a series of puppet emperors.

In the Byzantine era, Roman emperors and usurpers often continued to be drawn from the ranks of the army. There are ample examples of men being drawn from humble origins into positions of power, including Basil I, founder of the Macedonian dynasty and Romanos Lekapenos. It is however, worth noting that from the Kommnenian period of Byzantine history onwards, executive power was largely concentrated in several interlinked Constantinopolitan families. This is not to say that is was necessarily impossible for men of humble origin to achieve leadership roles in the military - only that they did not become emperors.

One of the most important takeaways should be that the Roman military evolved considerably over the nearly 1500 year span of the Roman empire. Recruitment practices varied during these times, as did the power politics of the day. While it certainly appears possible for soldiers from poor backgrounds to ascend to positions of power, the percentage of lowborn officers likely varied significantly depending on the period.