Can pet birds (parrots, canaries, etc) live in the same house with dogs and cats?

I am coming to believe that it isn’t cruel to keep birds in a “large enough” aviarium or enclosure, as long as its needs are met and it has such space. I also know that most who keep birds as hobbyists allow them some freedom in the house and they, like crated dogs, usually go back to their cages on their own—where they are secure and where their feed and water and comfort is. I even know that some bird hobbyists have other domestic animals in the house.

Howbeit, I cannot imagine pet birds and domestic cats coexisting. Though some say this is the arrangement.

Can birds be allowed freedom in the home if there are cats? Can one even KEEP cats and birds under the same roof?