I had two of my cats already when I got my first chinchilla. One of the cats was about four years old and the other was about eight years old. The oldest cat lets me know when one of the female chinchillas has given birth. She sits by the cage and just watches. The other couldn't care less about the chinchillas. He just wants love. After having chinchillas for about six months, another cat came into our lives. He is a trouble maker, but he tends to leave the chinchillas alone. (He likes to eat their hay, though!) Of course, I never let the chinchillas and the cats interact. When the chins are out of the cage, the cats are put into a separate room.
If you got a kitten who you raised around the chinchillas, he or she might be more likely to leave the chinchillas alone. However, I think it's critically important to remember that, if the cat tries to harm the chinchillas in any way, you cannot blame the cat. It is acting on its natural instincts.