Down syndrome is a chromosomal disorder than only affects humans, so cats cannot have it. There are other chromosomal disorders that affect cats and can have symptoms similar to those of Down syndrome.
According to the Merck Manual of Pet Health, many genetic abnormalities can affect cats. Some are just cosmetic, such as having extra toes or a calico coat pattern. Some affect the cat's health and may be fatal, such as ones that prevent the cat's metabolic system from functioning properly. Some of them affect coordination, facial structure and intelligence, which are similar to the effects of Down syndrome on humans.