It's not the ideal way to keep them. They prefer to live with at least some members of their own species, but a lone Cory will get along well enough with other kinds. I wouldn't suggest buying one of each, but if I see only one Cory of a species left in the lfs, I would buy it and it would have a better home than if it ended up in a tank with no other Cory friends.
The fish commonly called an emerald Cory is actually a brochis. It is related to and is very similar to corys but is somewhat larger.
Edit:
Here's a picture that shows at least 3 different species of corydoras swimming happily together the morning after they arrived at my house. While they might not have schooled together in the wild, they got along fine in my tank.