What Health Problems Do Purebred Cats Tend To Have?

Most purebreed cats have some genetic bottlenecking in their heritage.  This increases the chances of inherited disease and disorders, much more than if you get a mixed-breed moggie.  Feline hip displasia (where the hip joints are misaligned) is one of the most common problems (it also appears a lot in pure-bred dogs).  But that said, pure-bred cats by and large are much healthier than pure bred canines.

Specifics...
Siamese cats have a relatively high mortality rate among young kittens, and (some say) are increasingly neurotic in temperment.

Persians (and other long-furred cats) have the type of problems you would expect to have in an animal with such a high maintenance coat.  They are especially prone to furballs, something that the owner can help prevent with frequent grooming and even, occasional shearing.  As a breed, Perisans are prone to polycystic kidney disease, (which can be screened for), and a tendency in males to retain testicles.  The tendency to create very flat faces in Persians has also led to respiratory problems in some lines.