You should not have a caique and cockatiels in contact with each other. A caique has a much stronger beak and is likely to kill the cockatiels. These birds come from entirely different regions and would never encounter each other in the wild – they are not intended to co-exist other than being in the same room, in separate cages. They should not be able to get on each others cages because they can cause injury even through the bars of the cage.
I’m not sure if you meant in a room set up just for the birds, or in a room with people where they would be on a playstand or something similar. In a room around people, it is still best for them to take turns because birds can move fast and an attack can happen with little to no warning. If you are making a room for your birds, be careful about giving them free range. No matter how many fun bird safe toys and items you put in there, they are going to chew on the walls, molding, window seals, electrical outlets and lighting fixtures. Birds have a knack for getting into trouble – much like a small child. So keep this in mind when designing an environment. Keep in mind that cages are not a bad thing and if set up with fun toys and things your bird likes, the bird will view the cage as a safe place to be and a safe place to eat and sleep in. Play areas are fine, but they are not a substitute for their cage.
Thank you for asking Lafeber,