Unfortunately, as I well know, that isn't always true. My parents had to get rid of their cat when my older sister developed asthma that led to pneumonia when she was 4 years old. (Dad claims that it was a toss-up, but eventually they decided to keep the sister instead of the cat ... Once the cat was gone, her breathing problems cleared up considerably, but she is still asthmatic, and when she was tested for allergies as an adult, was told cats, dogs - anything with fur is off-limits for her.