I
don't believe you cannot tame and untamed bird. It might be a little harder to gain trust with an untamed bird but its possible, just have to put the time and patience into working with him.
I have been trying a slow approach with Yoda. I am completely letting him come to me, I'm not trying to pet him and hold him. Its a slow gain of trust. I think I'm almost there with Yoda, he let me touch his beak the other week without snapping at me. Its taken a year. Yoda loves preening me.. and talking to me

, but I can only touch him on his terms. It helps if you have something they want!
Jibby was much easier, he was tame when we got him, hes an odd one hehe, never had a problem with him, always been trusting of me, hes such a relaxed boy!
My 2 are out of the cage from sunrise to sunset unless I go out. They prefer to hang out together, flying about and playing, but they come to me when they prefer time apart or some quiet cuddle time. They also put themselves in their own cages for naps and bedtime. But they are always going in and out of the cages as they please. I have the cages right next to the computer so most of the day they are right next to me, unless they are playing in the bathroom together hehe. So I guess I interact with them all day. That will of course change when I get a job.