There were a large class of men who were equipped as "knights" in the stereotypical sense (heavy armor, lance, horse, etc.) Men-at-arms was a catch-all term for these sorts of men, who were typically seen as gentry. They were not as wealthy as knights often were (the increasing costs of being a knight were one of the factors of the emergence of men-at-arms as a category), but neither were they poor and downtrodden. The sheer fact that they could afford armor, weaponry, and horses meant that they were financially and socially a cut above the majority of the population. Some of them were relatively poor, as there are cases in England where former men-at-arms seem to have become archers due to a loss of income. However, being a man-at-arms could also be a tool of social mobility. A soldier who distinguished himself in the field might find himself knighted. Common soldiers also aspired to become men-at-arms and gathered equipment/saved money in order to afford the necessary gear. Men-at-arms might be attached to retinues or work for hire (either individually or in a free company of hired soldiers).

Focusing on the high (middle) and late Middle Ages (which is what ASOIAF is based on) and England (which is also what it is based on but, also, my knowledge area), the answer is yes and no. (ETA: The early Middle Ages was a much more fluid time.)

There were definitely knights who had little property or money to their name. They were usually younger sons of lower gentry. So, they were raised in a noble home but their older brother would inherit any property. And their family wouldn't necessarily be wealthy enough to provide for the younger sons. (Obviously, the Church was also a popular option for noble born younger sons.) These boys would often be sent out to foster at another noble home where they would be squires and learn to be knights. (To be clear this was also common for well off nobles. It was a common practice.) They would then be a retainer of a lord but they could move service from lord to lord over their careers. They'd have their equipment (which was valuable) but rely on Lords for their food and housing. The Blackfish fits the model of this a bit except he would have been retained by his brother if not for a falling out.

For the Norman English a retainer knight (not an official name given to them) might serve in England, Ireland, and through out the English French territories during their career. At least until the French territories were lost under John. Even then a knight without property of his own could serve through out Europe or in the crusader states.

A famous example of someone who began his career this way is William Marshall. Thomas Asbridge's The Greatest Knight is a very readable biography of him, if you are interested.

But, no, they don't really share all the characteristics of the hedge knights in ASOIAF. Maybe a little bit like Duncan the Tall (but even then not really). Or even Brienne (who serves different masters through out her career but is respectable.) They are people who would come from means relative to the vast majority of their society just not compared to the upper echelons. The profession wasn't necessarily looked down upon the way it is in ASOIAF. Individuals may be but as a class they weren't. Especially around the period of the Crusades (or the first few, at least) knights were highly esteemed even those who just barely made the rank. During the crusades there was also plenty of use for them. Plus, they would come from fairly respectable families, which mattered at the time.

The hedge knights in ASOIAF are often more like freelance mercenaries.

Edited to fix typos.

How about the "knight-errant" that is central in Don Quixote? Same idea as a hedge knight? Any basis in reality?

Expansion on the question: would hedge knights be more comparable to ronin, the masterless samurai of Japan?

Adding on another question about GRRM's hedge knights-

In Westeros, hedge knights are so common partly because any knight is allowed to knight whoever they see fit. Was there ever a real-world analog to this?

Fantastic response so far, particularly from /u/plinythehipster. To narrow things a bit, s/he is getting at what I'm most curious about - the social status and economics of a young man who would find himself in such a position. Obviously simply owning the armor and horse, not to mention having martial training, necessitates a certain level of opportunity.

A younger son from a modest family seems like a pretty reasonable scenario. I'm wondering precisely how common these sorts of individuals might be, and how well they match our romantic idea of the errant, down-on-his-luck knight who just needs a chance to prove his mettle. And furthermore, if such individuals existed in any significant number, what were their chances of achieving real social mobility?

Or, is it far more accurate to abandon that chivalrous image as fantasy? Is it more likely that a man who happened to have arms and armor is indistinguishable from his mercenary counterparts from other ages in which chivalry was not a social institution?

I always thought the hedge knights sounded a little like Ronin in feudal Japan. Can anyone speak to that?

Related question: Are hedge knights where we got the term "freelancer"?

Previous AskHistorians post about this: Were hedge knights a real thing?

Isn't GRRM's depiction of a Hedge Knight more closely based upon the Medieval romantic literary tradition of the "Knight-errant" such as Amadis of Gaul?