A viral example, herpes B, won't kill monkeys but it will kill humans. Aids kills humans, but not chimpanzees. Viruses are small and adaptable. While some viruses have little or no effect, others deplete systems and bring on disease. They can lie dormant for years, like HIV, then suddenly become deadly.
All viruses mutate, many times a year.
They figure out how best to thrive, adapt and move on. Again, just as visible species adapt to Earth by natural selection and evolution, so too do the microorganisms living in our bodies. However, as many have pointed out, you can't screen for disease agents that you don't know about. Pigs may contain viruses in their cells that have not yet been discovered and that do not affect them but may kill humans. Ian Wilmut, the scientist who cloned Dolly the Sheep, recently announced plans to stop experiments that would have eventually led to producing pigs for organ transplants.