Let's say there are lots of different countries, where mages are the ruling class (mostly). They aren't godlike and their abilities are mostly battle-related (there are some exceptions though). It's difficult for normal people to kill them, but possible. Mostly in cloak and dagger situations. In general there are a lot of similarities between our medieval nobility and mages. They're the main fighting force in the wars between different magocratic countries. Their abilities improve under heavy pressure like life-and-death situation, hence lots of battles and lethal accidents. A family member of a mage has a much higher chance to wake up their magic, but that doesn't mean that some random nobody can't.
Most countries have some sort of education institutes. But let's focus on a single powerful kingdom where two-tier system exists. There are those who were accepted to the education institute and those who were weeded out.
The former are the ruling class. They have access to teachers to learn more about their abilities. Their learning conditions are safer too. Lots magic-related knowledge exists in a written form, collected by the education institute and mages of this group. Studying magic takes years.
The latter have to learn on their own... either by stealing from the first group, fighting for knowledge against each other or developing their own spells (which can be dangerous). Some of them band together, others try to survive on their own. Unless they are very violent or incredibly damaging to the kingdom/king or some influential mage, there is no real action taken against them. Worst case, they have to deal with a revenge driven mage.
There are powerful mages in both tiers. The first-tier mages have access to teachers and written knowledge. There is little chance to die while studying. The second-tier mages progress through life-and-death situations in many battles against each other and the first-tier, thus their abilities escalate faster: spells get more powerful, they have access to more arcane energy. However, the second-tier mages can't really form a united force and therefore unable to fight against the first-tier mages as the whole. In a way it's a long, safe and rewarding vs fast, dangerous and not-really-rewarding.
An example (EDIT)
Becoming a mage changes a lot of things for normal people. Let's assume a young mage John from some poor magic-less farmer family wakes up his magic. In this world it's a known fact most of time. It can get quite violent and public. Besides, the Institute traces the waking accidents and issues notifications. It's possible to hide, the Institute knows the place, but is unaware of who "woke up". New mages are required by law to attend an interview at the Institution. It's possible to hide, but there's a very high chance to get caught.
John's family automatically becomes somewhat prominent. He and his family members will be highly sought as "breeding stock", because of their higher chance to birth children who have magic. Magic is a great chance to become rich and somewhat influential.
Then John fails the interview at the institution and thus doesn't have any protection offered by the kingdom. John's life gets a lot more complicated. If he does have some unusual abilities he may be sought for these abilities, pressured by those who want more power for themselves. He's no farmer anymore, he's a battle-mage. Would he want to work in a field being someone who can throw fireballs? Even if he wanted to be a farmer, he wouldn't really be able to due to pressure of those who would want to use him or his family for some reason. In a way it's easier for him to flee the country with his family instead of fighting for it in a war
Now... the questions:
Why wouldn't every mage be educated properly?
I have a few ideas, but unsure if plausible:
The kingdom has not enough resources to create safe environment for every single mage. Not enough teachers, not enough "money", not enough time.
Admission is based on character rather than abilities. Those who will obey the king/influential mages or those who would thrive in a safer environment are accepted, while those who are too selfish/individualistic... those who would have trouble to serve are rejected.
Is that reasonable enough? Is that too naive from political / human psychology angles? Can there be more reasons, perhaps?
Why would the second-tier mages participate in wars against other countries and fight for their country, which failed them in a way?
The price of being left alone (unless they cause too much trouble) is participation in battles against foreign mages. But why would they agree to fight? I have a single idea:
Families and friends. Most of them have a family or friends to protect, as a result - they fight.
I can't, however, imagine that every single second-tier mage would care about their family or friends enough to risk their lives fighting for their country in wars.
They're mercs paid for fighting. Money or knowledge.
Reasonable? Are there some other good reasons for them to participate and hence being left alone despite crimes?