I've read in some places that there really isn't a specific number of traits they can have (I guess you could possibly attain all the hidden traits if you wanted too). The main thing is the traits they have been trained to do can be lost if you don't keep up with it. You can only give your pet 3 traits in CAP and the rest you have to train them.
If you scroll down towards the middle of the page it gives you all the traits you can get for your specific pet. [Link removed. - Pam]