Can animals also get Down syndrome?
Animals can be generated with genetic defects similar to Down Syndrome, but not that exact condition, except in the case of great apes.

Down Syndrome is a kind of defect called a chromosomal abnormality, meaning that either there is an extra chromosome or an excessive repetition of the same genes on a particular chromosome. In the case of Down Syndrome, that repetition in on a region of the 21st chromosome. Similar abnormalities in other chromosomes are possible, such as the abnormality which causes Klinefelter Syndrome.

Although animals do have chromosomes, they have different chromosomes than human beings do. For example, a horse has 32 chromosomes whereas a human has 23. Apes are most similar to humans, having 24 chromosomes. Therefore, if trisomy is to be found, apes would be most likely to have it. Researchers have attempted to answer this exact question and there is a paper on the subject entitled "Conservation of the Down syndrome critical region in humans and great apes".

The result of this study was that the area on chromosome 21 in humans that is responsible for trisomy is found analogously on chromosome 22 in apes and that apes can, indeed, get Down Syndrome, if they have duplication of this region. There do appear to be apes that suffer from this condition. The band diagram for both chromosomes is shown below:



where "H" indicates human and "C" indicates chimpanzee.