It depends on where and especially when. Traditionally not really because a feudal society implies a somewhat rigid hierarchy.

But that doesn't mean you can't climb up, what change is how much you can climb: In Western Europe most people were serfs in the Middle Ages, meaning they were legally bound to the ground. The literally were forbidden from leaving their village (this was true in Russia up until the late 19th century by the way). Now depending on when and where you are it was absolutely possible to leave serfdom (usually buying your freedom). You will after that be a freeman who would usually rent your land. That's a step up. So there is definitely room for progress here.

What is important to understand is that there a feudal society is not simply "lords rule over peasants, and that's it". There is an internal hierarchy that you could climb: in a village you had poor serfs, you had free men tenant, you had even somewhat rich farmers. In cities you had citizen/burghers, people with specific rights (economic and political) in the city (that's usually the merchants and tradesmen) who were above the people simply living here.

So yeah, you have a complex social structure which you can absolutely climb with luck, connection, marriage, so on. The one barrier that is usually impossible to cross is the nobility: the nobles rule, they are on the top of the pyramid. And if you're not a noble, you're not getting in. In the "classic" feudal system, you cannot cross that. You have a whole internal hierarchy that goes relatively up, but it stops short of actual political power. This changes later, in later centuries during the modern period (15-18th century) in France specifically you could buy a noble title, but it was incredibly expensive.

So could you rise up? Usually no. If you could it was very local. The main thing people could hope for was no longer being serfs. For people in town trying to get into the craftsmen class. So social climbing, but far from rags to riches.