This absolutely happens in nearly every part of our daily life. Its almost impossible to go through a day without running into it.
The tricky part is realizing what "leveling up" must mean. Leveling up must be a change in state which can be captured numerically. Given that it's not magical, it must be a change in perceived state, showing how society views you and your skill level.
We use this all the time. If you drive to work, you have a driver's license. Most of the time, we keep track of our own driving skill informally. However, when we get pulled over, we have to produce this license to prove that we have "leveled up" enough in driving skill for the government to grant us the right to drive. This involved a written test, a driving test, and proof that you hadn't gotten in any DUIs and lost your privilege since then. This is functionally indistinguishable from "leveling up."
Virtually all of our first-pass estimations of people's abilities stem from observing their level. If you want to go get an engineering job, you don't get there by showing prowess in engineering. You get there by showing a piece of paper from a good Engineering college known as a diploma. You might even have to show your GPA. Want to enter the IT field? I don't care how good you are at managing Cisco routers, you won't find a job unless you have attended the proper courses and gotten your Cisco certification. Even the lowest paying entry level jobs at least want some proof that you've leveled up enough to have an address (and a shower you can clean up in).
This process is natural. Many skills take quite a lot of time and energy to observe properly. We can't waste that much energy, so we assign symbols to signify the exertion.
The drawback to the process is that you have to presume someone knows exactly what skill you have sufficiently well to test you. Arts in particular are notoriously difficult to pin down with a test. There's just too much subjectivity. You also have to be careful to avoid people teaching to the tests. In America, we've been pushing standardized testing for years as an effective way to measure the progress of students. The result has been a laughable failure. Students are entering college and complaining that they are woefully unprepared for college life because their entire school life, from elementary to high school, has been centered around how to pass the standardized tests.
My favorite compromise is the kyu/dan system used in Japan for Go and many martial arts. You also see mirrors of it in martial arts belt colors. One starts at a high numbered kyu(like 30 kyu for Go), and progress towards lower numbers. Eventually you get to 1 kyu. The next transition is from 1 kyu to 1 dan, then 2 dan, 3 dan, and so forth (typically there are 3-7 levels of dan). The belt analogue is that the colored belts are like kyu ranks, and the different degrees of black belt are the dan ranks.
Many of us assume a black belt in karate means "you know karate." However, that is a bit of a misnomer. A more accurate phrasing would be that a black belt, or a 1 dan ranking, indicates that you are "minimally competent in your art." You know just enough of the basics to claim competency. Black belt is not the end of the road; its the beginning of one.
There is a distinctive shift in the way the testing or "leveling up" process works when one shifts from kyu to dan. To achieve a new kyu rank, the primary focus is an objective measure of your ability to do some skills. In theory, a 1 kyu knows all of the objective skills from the art. Once you enter the dan ranks, the tests become more subjective. To become a black belt/1 dan in many martial arts, one has some subjective portion of their testing. In my Tae Kwon Do class, a black belt was expected to produce an essay "What does a black belt in Tae Kwon Do mean to me." There is no right answer to this; each essay is different.
This shift lends itself to a nice mix of the RPG style "leveling up" with a buffer to permit more expressive arts. At first one is expected to build up basic skills, a. la. leveling up. After that, however, it is recognized that no one size fits all. You have to find your own soul in your work, and you are credited with a 2 dan or a 3 dan rank when the community believes your expression of the art warrants a new level.
With this, you also see a natural check on liars cheats and thieves. At the low levels, if a yellow belt decides to buy an orange belt to sneak into some training, minimal harm is done. On the other hand, if a charlatan buys himself a 3rd degree black belt and tries to open up a martial arts studio, people will quickly check up on his lineage, and spot his failure.
More interestingly, fellow black belts will detect the charade instantaneously. By the time one becomes a black belt, one certainly has the skill and experience to determine whether a fellow practitioner has earned their rank. Thus, if you know a black belt who is teaching, you might ask them to go say 'hi' to the newcomer. They will quickly tell you whether the newcomer is legitimate or not, with or without an official trip to the records room to find his certifications.