The humans call Jewel by her name (Jewel), although they would have no idea that this is actually her name. Her family, (who we meet in Rio 2), also call her Jewel, so she didn't take on the name after the humans gave it to her.
When the bird sanctuary guard rips off his clothes and begins dancing, his wrists are bare, but when he is giving his report to the police officer, he has fluffy straps around his wrists, matching the ones he has on his ankles.
In the beginning of the movie, during the first performance of 'Real in Rio' the yellow Macaw that baby Blu looks to has three chicks. She only throws two out of the nest then flies away with the newly flying chicks, leaving the third one in the nest.
When Blu is standing on the makeshift light runway preparing an attempt to fly, the section of wire at the end of the runway is shown draped and absent of lights, but as he runs the wire changes, it is no longer draped and has no gap in the lights.
When Fernando is carrying the birds to the smugglers safe house, Jewels rips off a hole in the cloth the cage is covered with to see where they are taken. She is situated at the front part of the cage and Blue is at the rear end - closer to the door. They pretend they are dead and lay still on the cage floor. When they come in, Fernando puts the cage down on the desk still covered with the cloth with the hole (Jewels had ripped) facing Marcel - the lead smuggler. Marcel stands up, goes around his desk and pulls the cloth off of the cage, facing the cage at the side with the door. He sees the "dead" birds and in this frame Jewels is next to the door of the cage - the birds switched places.
Toward the end of the film when Tipa tries to avoid the oncoming parade floats with the plane, he pulls back on the flight controls to take off. The scene switches to an outside shot of the plane's wing where the flap moves to the down position, but pulling back on the controls manipulates the elevator, on the tail, which would cause it to pivot upward. The flaps are controlled separately.
Standard Brazilian license plates are light gray with black letters. They are composed by three letters followed by four letters (i.e. ABC-1234) as opposed to the three letters/three numbers shown in the movie.
Flight is instinctive to a macaw, and a macaw is going to take off in fear if it sees one of the commonplace predators of Minnesota, feline or canine, likely to make a meal of such a bird should one ever find its way into the bookstore.
Sulphur crested cockatoos are native only to Australia an neighboring islands, and there is no similar South American species. However Nigel is a house pet and it is plausible that the smugglers have operated in other continents.