Insanity, used here to mean any sort of mental imbalance, is rare amongst a society like the clans. This is because the cats are taught from a very young age that they are expected to serve the clan rather than themselves.
This expectation of service means that self-oriented mental states such as depression and clinical anxiety are very rare. When you’re told from the first moment than your whole existence is for the clan it is very easy to cease dwelling on yourself and move forward. I am aware that in modern, western human culture things are not so, but this stems from a combination of genetics (those with imbalances don’t procreate in the clans and frequently don’t even remain clan members) and society (we consider personal happiness a right, while the clan cats have no societal or personal expectation of any level of personal happiness or fulfillment).
While this clearly outlines why behaviorally oriented problems might be rare, chemical problems do need to be addressed as they can exist.
Chemical problems are, as mentioned before, rare due to the lack of strong genetic influences that result in them. Such genes are rarely retained in the population, and are even more rarely bred within the population. Cats who display mental imbalances are not frequently considered suitable as breeding stock. Keep in mind, though, that it is only the extreme cases of mental imbalance that are rare to the brink of impossibility. Only those cases in which the functions of the cat within the society are impaired result in the cat being ejected from the clan or restricted within it.
Anger and impulse control problems, while highly uncommon, are possible assuming that the problems are not far-reaching (limited to specific types of situation) and do not interfere overmuch with the cat in society (examples of such cats: Bloodstar & Flamesong). Certain levels of paranoia are also possible given the right circumstances setting them up (example of such a cat: Yellowfrost). Another example of a form of insanity that is rare but possible is very specific forms of suicidal behavior. Most specifically, things like starving oneself to feed others are examples of this; a clan cat who exhibits a more self-serving reason for suicide is unlikely (example of such a cat: Ravenwing).
Playing a clan cat with some form of instability is a balancing act in which you’re constantly considering the disability, the world they grew up in, and the tolerance of the leadership of that clan. Few situations exist in which a cat who is deemed dangerous would be kept in a clan. From first breath the cats are taught to believe that clan comes before all else. This means if a cat shows a high potential to cause harm to clanmates, the leader will not hesitate to make the choice between the good of the clan and that individual. And depending on the situation and the leader in question, the results could range all the way from exile to execution.