Yes. Diseases like AIDS are thought to have originated in primates and jumped to humans. One of the big worries is influneza from Birds or Swine jumping to humans as well. But getting an animal borne disease isnt’ always bad. If you think back to your Irish and English poetry its sure to feature a Pretty Milkmaid. The reaosn for this is that milkmaids would get Coxpox from cows and were then immune to smallpox, a horrible often fatal and disfiguring disease. Edward Jenner noticed this trend and developed a vaccine fro smallpox based on cowpox in the late 1700s. Smallpox is now errdicated from the planet thanks to cowpox!
Diseases passed from animals to humans are called zoonoses. There are more than three dozen we can catch directly through touch and more than four dozen that result from bites. For instance, swine flu, Plague is carried by rodents and even cats, and when transmitted to us it becomes most deadly, Malaria is transmitted through mosquitoes, rabies. There are many diseases which can be transmitted from animals to us.