Yes, cats and birds can live together harmoniously in the same house, as long as their owners are willing to take the precautions that are necessary for the safety of both.
If you have a cat and want a bird or vice versa, you should understand that in nature, the two would be enemies -- predator and prey. While the natural roles of your pets can be altered, it's important that both are kept under close supervision to ensure that no disputes arise.
If your cat shows an interest in hunting your bird, try giving her a quick spritz with a water bottle. Most of the time, this will be enough to curb a cat's predatory instincts after a while. Likewise, if your bird shows aggression to your cat, discipline is in order. Birds are able to catch certain diseases from cats, and direct exposure to a cat's fur (through biting, for example) can heighten your feathered friend's risk of contracting an illness. For this reason, and many more, you should try to avoid any physical contact between a bird and a cat if at all possible.
With consistent training, both pets should learn to tolerate each other rather quickly. Many owners have even been surprised to find that their birds and cats quite enjoy each other's company under proper supervision -- and that's really something to squawk about!