In a word, no. HBO's Rome suffers pretty severely in their depiction of social structure within Rome. From their insistence on having lower-class characters speak a different form of English than the nobility to this, social structure is definitely where Rome is most inaccurate.

In the 1st Century, BC, Roman social classes had broken down and really only existed in name (by the Principate they didn't exist at all, except in family traditions). Roman social classes were divided not according to wealth or land, but political rights, with the old patricians holding the power to hold magisterial office. By the 1st Century the successful campaigns of the populares under the Gracchi and Marius had either enacted legislation to grant voting and official rights to the plebs or had forced the more conservative factions to propose such legislation (for example, most of the rights given to plebeians were the work of Sulla). By Caesar's time therefore, all social classes retained the same official rights, more or less, and the only thing separating them was family history. Family history divided the nobiles, families which could claim an ancestor they had held the consulate, from everybody else, especially novi homines. But that was totally independent of wealth or education--most novi homines were quite wealthy and a large number of nobiles (who could be from any social rank, and many were plebeians) were quite poor. Caesar's major supporters were in a large part old established family members who had squandered their fortunes or whose families had lost prominence. And consider this: most important members of the senate and the government during this period were not patricians. Cicero, Marius, Antony, Crassus, Pompey--all of these men were not patricians, although several were nobiles.

Beyond political rights there wasn't much difference between classes. Wealth was not restricted in any way, nor were landowning rights. Different social classes frequently inhabited the same neighborhoods--Caesae, for example, who was descended from among the most ancient of the nobiles, lived for quite some time in the Subura, a district of the city known for its nightlife and poverty. By the time Cicero was killed he was among the most wealthy and influential men of his time, and he was a mere equestrian. Crassus' family, which was the first plebeian family ever to hold consular office, was incredibly wealthy and there was even a branch of the family with the cognomen Dives, rich, because of the famed wealth of one ancestor (this was not Crassus' branch. Although Crassus was wealthy, his lack of military holdings and gubernatorial positions prevented him from reaping the main money-maker for Roman politicians--tribute. Because of this Pompey was actually unimaginably more wealthy than Crassus). Pompey was, until Caesar took control of the state, probably the richest man in Roman history, despite being an equestrian, and had the resources of every city from Pontus to Judaea at his feet. Indeed, even Pullo couldn't possibly have been discriminated against on these grounds, since as a primus plus he would have drawn a quite enormous salary, several hundred times as large as that of a regular soldier and dozens of times the salary of a junior centurion.

If there were any grounds for social discrimination it would've been education. This obviously was heavily dependent on wealth, but particularly wealthy people who were uneducated would've been perceived poorly. This often occurred later in ran history, as rich foreigners became more common than Romans or old noble families that retained their wealth, but it was present already. We see this in the Satyricon very vividly, but it's in other places as well. However, this is not something that would keep you out of public establishments, only private parties and stuff. Moreover, it's not something that someone like Pullo could be accused of. Centurions had to be able to read and write and had to have a minimum amount of organizational ability and so forth to advance. In point of fact, centurions tended to be quite decently educated, and the primus pilus would have been among the most distinguished and best educated men in the legion, even if he hadn't necessarily received as much training as a legate.