I would think a Sun Conure would have a go at a cat. When a cat and bird are raised together it is possible to let them interact without too much trouble - that is what I have heard, however I would not like to try it. Some bigger birds are not seen as prey by cats but anything smaller could get into trouble. Personally, I would not trust any cat in the same room as a bird. It takes only a moment's inattention for a cat to be able to either kill or seriously injure a bird.
I would be wary of a cat, its too much of a natural instinct. I have a Hahns and a blind dog, i have to be careful of, as my blind girlie just comes up very close and cant see how close she is. My Hahns is very good, but a bump, or nose too close may startle her.
My Green Cheek Conure ATTACKS my cat. In order for an animal to live in this house, they all need to be able to out at the same time. Rosco and CAT are never left together unattended, nor do I allow them to interact without me holding one or the other. CAT won't go to close to Rosco and if Rosco feels uncomfortable, he charges CAT.
It's all about the individual animals. I have mellow dogs and a kitten. If I raise CAT now to be around Rosco, then I don't see a problem. The dogs are gentle and I trust them completely.