Is it possible to have twin birds?

Baloo mentioned fraternal twins, and sometimes there are eggs with two or more (record is 9, see below) yolks, which if they developed right would be identical twins/triplets/etc. I can't find anything saying that they can actually develop right though.

http://www.madsci.org/posts/archives/dec97/879963549.Gb.r.html
"In a chicken, fraternal twins are known to us as two complete eggs with seperate shells. Identical twins, on the other hand, come out as one egg shell with two yolks inside. "

Yes, it is possible, but rare.

When two chicks hatch from the same egg, the egg usually has two yolks. Usually, one embryo out competes the other and only one chick survives to hatch. Many time both embryos die before hatch. We have no knowledge of Siamese twin chicks ever being hatched. The development of twin chicks from a single-yolked egg.

http://extension.psu.edu/4-h/projects/poultry/embryology/teacher-resources/faqs

And here is an actual occurence:

Roger Olver watched on as the chicks emerged to become the first recorded twin ducklings to ever hatch and survive in Britain.

The ducklings - named Romulus and Remus - are to be spared the table and become pets at the Cornish Duck Company in St Austell, Cornwall.

http://www.telegraph.co.uk/news/uknews/6431011/Twin-ducks-hatch-from-one-egg.html