Yes!

The idea is born out of the notion that a person with a stake in the polis, should defend the polis within his means. The Roman Republic, with its tribes and other administrative formations, is not unique in this principle; many city-states in Greece followed similar principles; a man was expected to have a stake in the state in order to serve. Naturally he provided his own equipment. This was far from the exception to the rule at the time, or even in the past. Rome would reform to a professional military later in the Republic, but for the majority of its Republican existence it behaved similarly to other such city-states.

This was a very common notion. Ancient war focused on farmland. Armies would draw each other out by attacking crops and threatening harvests. The goal would be to eventually get at the city and capture spoils or slaves from the local population.

That means the people with the most to lose were the farmers. They needed to fight to protect their income, and the city supported them because they needed the produce to support the urban population.

Added to this was the whole notion of honor. In Rome they had the opinion that being in the military was a way of showing status. They were also so successful that a soldier was guaranteed to come out of the army wealthier than he entered (well, that was probably an exaggeration on the level of claims made about emigrating to the US in the early 20th century). Therefore they felt obliged to limit entrance into the army only to people who deserved it, the yeoman farmers who represented the ideals of Rome.

This changed under Marius who enrolled the landless poor in order to gain political power. Because being in the army was seen as such a great thing financially and for your reputation, Marius could gain a political following by offering it to the urban poor who had become more and more numerous and enfranchised (I don't buy the consensus idea that Rome had a shortage of farmers from which to levy their armies).

A lot of Rome's peers did not have the same attitude. Carthage was almost entirely a mercenary army. That led to a huge war after they were unable to pay the mercs after the loss of the First Punic War (the Mercenary War was far more destructive than they first two Punic Wars). Roman strategy in the Second Punic War focused on weakening Hannibal's army by disrupting his supply of money from Spain. They sent legions to Spain even though there was a hostile army in Italy that had already defeated three Roman armies! And the strategy worked!

Greek states began to supplement their armies with Mercenaries during the Peloponnesian War (Athens captured 200 or so Spartans on Sphacteria with a force of Aetolian mercenary peltasts). Persian and Hellenistic monarchs used mercenaries for most of their armies. Professional soldiers were just better than farmers, and the nature of war had changed from defending your own property to projecting power on borders.

Rome also relied on non-Roman units. Each legion was typically supported by an equal force of auxilia, non-citizen allies. They weren't exactly mercenaries because they served under treaty, not for specific payment, but they aren't yeoman soldiers either.

For more: Victor Hansen, Western Way of War. Nate Rosenstein: Rome at War.