Nobility is something that differs from country to country, from period to period. Maybe you have a specific country or region in mind, but I will talk about the French nobility during the modern Ancien Régime and the constant risk of derogating from the essence of nobility.
What distinguishes a noble from another man is the work he does. A noble is to live off his lands thanks to rents or service to the prince/the State in the army or in the parliaments for examples. To live nobly. As such, performing other jobs is somewhat un-noble, too similar to what the common man does. And then there is a risk of derogating from the nobility (déroger à la noblesse) which mostly meant not losing lands or titles (not all nobles had titles or lands) but losing the privileges granted to the nobility. These privileges are mostly financial and nobody wants to pay more taxes like a commoner does such as the taille or being unable to buy a military office. The most controversial point concerned trade and whether or not a noble could become a merchant was a debate in the 17th and 18th centuries. In the end, they never were allowed to be merchants but some tried, and some were caught.
Although some theorists say that nobility can never be stripped because it is in the blood (Gilles André de la Roques writes in the 17th century that the power of the nobility comes from the seed and the virtues of the ancestors, from the race.), in a legal point of view it can be taken away from an individual. We know it was done quite frequently because the monarchy saw in these derogations from the nobility a way to make money. Selling patents of nobility is a good way for the monarchy to make money, a lot of money because many people, especially in the mercantile or industrial, bourgeoisie wish to be exempted from all these taxes and gain rank in the society. So the monarchy made these rules about derogating from the nobility but also made patents to gain it back in exchange for money. Some sort fine in fact. These were called patents of rehabilitation. An example was the Riquet family, barons of Bonrepos. It is sometime during the 16th century that the family derogates from the nobility. As far as I know, we do not know why maybe because of working their fields themselves out of financial necessity but Pierre-Paul Riquet obtained a patent of rehabilitation in the early 17th century.
You may be aware that France during the Ancien Regime is very complicated regarding its laws and administration. There are exceptions to every rule, even exceptions to the exception. Individual or regional exceptions make it very hard to define a general rule but in order to avoid having an empty treasury because of privileges or too many idle nobles getting poorer because they cannot work is a constant preoccupation of the monarchy. As such some jobs, even manual labors were allowed such as glass-blower, slave trade, trade not involving owning a shop (there is what is called a merchant nobility encouraged by the monarchy during the 18th century to help the economic growth), ...etc.
There are other examples of course, but rarer. During the trial which sentenced him to live the rest of his life in prison, Nicolas Fouquet (1615-1680 ) was deprived of his nobility and most of his possessions because the crime he was accused of, manipulation and maladministration of the State Treasury was since the 16th century punishable by the loss of nobility. But it is personal and does not concern his family and children.
In short yes being un-noble was something happening in France because being a noble supposed a certain behaviour and specific duties and conditions as well. Derogating from the nobility, meaning ceasing to live as a noble, was the most common way to lose one's nobility. But it is not necessarily permanent if the concerned individual has enough money and can convince the king or the chief of heraldry (juge d'armes du roi) to give him a letter of rehabilitation. Then they are exceptional cases such as Fouquet, but they are rare at least during the modern period. If a patent of nobility can be given it can be taken away as well.
Source:
Gilles-André de la Roque, Traité de la noblesse, de ses différentes espéces, de son origine, du gentilhomme de nom et d'armes, des bannerets, des bacheliers, des ecuyers, & de leurs différences .., 1710 (published).
François Bluche, La noblesse française au XVIIIe siècle, 1995.
Alain Texier, Qu'est-ce que la noblesse?, 1988.