I have always had cats – and for the better part of my life I have also had pet dogs and birds.
I have taken in stray cats that were used to killing birds and mice for food; and yet it took me only a couple of weeks to a month to train the cats to leave my indoor birds alone.
I have had several large dogs (Doberman, German Shepherd, several mixed breeds, as well as one American Eskimo, etc.) and they were perfectly fine with my birds -- after careful introduction and training. I am not a very patient person and training does not come naturally with me. If I can do it, anybody can. All it takes is PERSISTENCE!
This being said -- some breeds are more unpredictable than others! Therefore, good judgment in assessing your pets is of the essence.

The biggest mistake people make is to try to keep birds and cats / dogs apart (in different rooms or relying on a cage to keep the bird safe). Eventually someone is going to forget to close the door properly, the bird gets out of his cage somehow, and the cat or the dog is going to take advantage of that. Eventually this is going to happen .