Dogs cannot have Down syndrome because it is a genetic condition caused by an extra copy of human chromosome 21, and dogs have a different number of chromosomes (78 in total, in 39 pairs). There is no equivalent to human chromosome 21 in dogs, so the chromosomal abnormality that causes Down syndrome in people does not occur in dogs.
However, dogs can have other genetic or developmental disorders that cause symptoms resembling some traits of Down syndrome in humans. These include congenital hypothyroidism, pituitary dwarfism, congenital hydrocephalus, growth hormone deficiency, or other chromosomal abnormalities that may manifest as cognitive delays, physical abnormalities, or developmental delays.
In summary, while dogs do not have Down syndrome, they can suffer from certain health conditions that may mimic aspects of it, and any such symptoms should be evaluated by a veterinarian for proper diagnosis and care.