I had asthma when i was a toddler and we had a cat called tigger
my mums baby hehe
Though the cat was in contact with me and it may of annoyed my asthma my mum never thought about getting rid! she was family!
My dad has a mild form of asthma and didnt realise this until tigger passed away and he never had it again,only as he goes into his 50's is it coming back.
I occastionally get moments where im breathless or weezing but thats usualy cold weather or im becoming ill. The cats though never bother me.
Dawn also has ashma which can at times cause her problems but weve had cats for over 2 years now and never have we found it impossible to be around them.
I know were quite lucky to have a mild form of ashma and there are stronger cases that can be life threatening which then i would understand rehoming an animal,but surely in most cases you can put up with an inhaler to make you breath easy for the love of a purring kitty cat?