If your cat is an active hunter, bringing in a chinchilla may not work out well. Even when the chinchilla is in the cage, a determined cat could terrorize a chinchilla by reasing through tthe cage bars or even just staring at the chinchilla. After all, chinchillas are rodents and a cat that likes to hunt is likely to consider your chinchilla fair prey.
If your cat is more laid back or mellow, or doesn't care much for hunting, then you have a lot better chance of the cat pretty much ignoring the chinchilla, and the chinchilla will likely get used to the cats presence and not be overly stressed.
That said, I recommend close supervision of cats around the cage (of any rodent) for the first while until it can be determined that the cat isn't going to go after the chinchilla. When the chinchilla has free time out of the cage, the cat should be put in another room or restrained (at least for the first while). And, even with the mellowest of cats, never leave a chinchilla out of the cage unsupervised with a cat.
Of course, if you have the space and time, you can make it work even with a hunting cat by always keeping the chinchilla and cat separated (obviously, this requires a lot more care and juggling of pets' needs).