Oh look, I get to geek out about the Pastons again!

In medieval England, at least, it was possible - although not common - to move up both financially and in social class. (The two things were definitely not synonymous, and still aren't in many societies.) One example is the Paston family, who totally rule because they wrote each other letters and kept them for centuries, so we know a lot more about them than we do about most families.

Around 1400, we've got Clement Paston, a Norfolk yeoman or possibly even a bondman. This account was written well after he died, by someone who was dissing the Pastons as social climbers, but still, it's clear that these were not people from a high social class:

First, there was one Clement Paston dwelling in Paston, and he was a good, plain husband [husbandman], and lived upon his land that he had in Paston... The said Clement yede [went] at one plough both winter and summer, and he rode to mill on the bare horseback with his corn under him... And he wedded Geoffrey of Somerton's sister, which was a bondwoman.

Clement, though, had plans. He borrowed money and relied on a brother-in-law's help to send his son William to be educated. William became a lawyer and eventually a Justice of the Common Pleas (dealing with civil matters between two individuals), made plenty of money and bought a ton of land including Gresham Castle, and moved another rung up the class ladder by marrying Agnes Barry, daughter of Sir Edmund Barry. In one generation, the Pastons had gone from one-plough farmers to castle-owning members of the squirearchy. No inheritance involved.

Edit because I left out something that's equally relevant:

William's son John kicked things up another notch. He was also a lawyer, and he made friends with the Norfolk knight Sir John Fastolf, who ended up leaving the bulk of his considerable property to Paston. Fastolff's previous heirs were understandably not pleased: this led to decades of fighting, both the legal kind and the armed kind, but the Pastons eventually came out of it with Fastolf's Caister Castle. This put them in the big leagues. While inheritance was involved in this step of their rise, it wasn't the bloodline-based inheritance your question implies; it was due to Paston's networking.