Is it possible to find data which makes sense when decrypted with different keys? In theory, does a piece of data exist which, when encrypted (using any method) with a given key, makes some sense when decrypted with the wrong key? 

Here, a definition of "some sense" might be something like "when interpreted as ASCII character codes, the data forms a series of mostly correctly spelled English words". In essence, a definition of "some sense" would be that the decrypted data looks (almost) like it was correctly decrypted.
Yes. How to find such data depends on which encryption you are using. With one time pads it's trivial: You can freely choose another pad that when used as a key will decrypt the ciphertext to an arbitrary message (of proper length).