Not a cat biting its own tail but you can see the association - photo by Malingering (Flickr)
The cat biting tail problem is not that unusual (and I am talking about a cat biting its own tail, by the way). It looks like the cat needs a psychiatrist. In fact quite a few people think cats are a little mad to go around chasing and biting their own tail as if it is prey. Some cats do it so purposefully that they draw blood or cause themselves pain. I don't think they are mad at all but the causes are not totally understood.
The most likely reason is a psychological one but one that is also perhaps logical and not irrational. There are a number of videos on the internet presenting the cat biting tail show and my first instinct is that the cat is simply entertaining itself but in a kind of obsessive way.
We are prone to do this sort of thing to. People sometimes pick at themselves to the point where they bleed for instance. I think is is a kind of obsessive compulsive disorder or displacement behavior linked to the condition feline hyperesthesia. Feline hyperesthesia is compulsive grooming (licking themselves too much) to the point were a cat developed bald patches.
A cat biting its own tail is play acting hunting. Playing at catching prey is very normal behavior. For confined and bored cats this natural desire to express hard wired instincts can, if the cat is stressed for whatever reason, present itself in slightly unnatural ways.
There also may be an underlying problem with the tail that might confuse diagnosis. Nerves damaged in an accident that we are not aware of can paralyse the tip of the tail (or the whole tail), for example. The typical accident is one where a car runs over the tail as the cat tries to escape. The tail becomes a dead object to the cat that is suitable to chase and chew on. Also cats are not self-conscious (aware of oneself as an individual or of one's own being, actions, or thoughts). Under these circumstances it is perfectly normal for a cat to chase and bite its tail if it has no other means to exercise its natural urge to hunt.
Another possibility is an irritation in the tail. Although the behavior of the cat in this instance would not be one of chasing and biting but of over-grooming and perhaps biting (nibbling) to. If the irritation is intense cats can jump into the grooming position quite quickly giving the impression that it is playing at hunting or needs to see a psychiatrist. The things that can cause irritation might be cat ringworm or mites for example. One visitor suggested blocked anal glands. I am not sure if that is true.
So, if our cat is doing the crazy cat biting tail game what can we do? See a vet of course but I expect that he or she might not have a ready answer. The kind of answer might be creating distractions or administering tranquillizers or even as a last resort amputating the tail. One cat owner reluctantly had this done and the cat was much more content indicating a kind of obsessive compulsive disorder. Certainly where the cat's tail is permanently paralyzed the treatment would seem to be amputation (Book 1).
I favour a more natural approach if possible. Remove or separate out all possible causes of stress (noise, another cat) and allow the cat the opportunity to express its innate character by making its environment as natural as possible. This means plenty of simulated play/hunts/stalking etc requiring our input or (and I keep saying this) a large cat enclosure (if facilities allow) for the cat to play outside in safety. I would also ensure that the food is as natural as possible. Cat food can cause allergies, especially modern commercial food and that can cause skin irritation. Add that to boredom and you might get a certain amount of what seems like unnatural behavior.
I would doubt very much that wild cats obsessively chase and bite their own tail. They are far to busy surviving chasing and biting the real thing. Cat biting tail syndrome is particular to the domestic cat; in which case it is essentially our problem as we create the environment, which causes it to happen.
See photographer's Flickr home page: Malingering