We've found abandoned broken eggs in our yard before. A few weeks ago Rob watched a brown-headed cowbird fly from the pine tree out back and drop a robin's egg just inches from our deck. Cowbirds are known as "brood parasites" that search out other bird's nests to lay their eggs in. They even go so far as to find nests with speckled eggs (to match their own so the host won't recognize them as foreign), and they essentially stake out these nests waiting for the hosts to leave before laying their own egg. Usually they will remove one or more of the host's eggs, perhaps so that bird won't notice that there's more eggs than she laid herself. I don't know if birds can count or have a sense for how many eggs they're incubating but it's some fascinating stuff!
The host bird raises these cowbird babies as their own. Last summer I saw evidence of this foster-parenting when a male cardinal was feeding a young cowbird. What really intrigues me is that these cowbirds are not raised by their kind, so how do they all know to do this when they mature? It must all be hard-wired into their little birds brains. It's simply amazing!