Yes, unfortunately for the birds, most people think of birds as 'low maintenance' but this is nothing more than a fallacy spread by the industry ("Give it a nice cage, put toys in it, feed it pellets and water"). So, yes, lots of birds out there belong to people who work full time and most of them will tell you that their birds are fine - and most of them will be telling you the truth UNTIL the bird is no longer 'fine' because it's overly hormonal and in constant pain from overgrown gonads. This doesn't happen overnight, it takes a while because babies and juveniles don't produce sexual hormones so the bird needs to become sexually mature first before it begins to suffer from a breeding season that never ends. It's the same thing with diet... you have lots of people telling you their bird is fine been free-fed protein food until the bird starts showing symptoms of liver disease after years of a bad diet.
Every single bird that is given up because of screams, bites, plucking, self-mutilation, etc is a bird that was kept under conditions that were not good for it. Nature doesn't make mistakes and thinking that we, humans, know better than her is not wise. Please research avian photoperiodism and you will see.
As a separate note, I have a female GCC (one of the pyrrhura species that Lorius was mentioning) bonded to an older, wild-caught, severely handicapped male peach-front (which is, actually, an aratinga species but small like a pyrrhura) and she is as bonded to me as any 'only' bird. She never bites or even nips, she doesn't scream or pluck or anything - she is the sweetest little thing! She happily steps up to my finger just to quickly climb up my arm all the way to my shoulder so she can reach my cheek to give me a kiss and, if I give her a kiss on her head or if I tell her "I love you", she kisses me again so, sometimes, we go quite a few minutes with a kissing back and forth: birdy SMACK - human SMACK - birdy SMACK - human SMACK
She then hooks her left foot on the neck of my top and putting her right on my chest, hangs there looking down at everything I do (I do all my morning chores with her hanging from my front). So, yes, it is entirely possible to have two birds bonded to each other and still have the close relationship of a bird bonded to you.