Do key collisions for symmetric ciphers exist?
The number of possible permutations of a block cipher are $2^n!$ where $n$ is the block size. A permutation maps all $2^n$ possible input blocks to $2^n$ possible output blocks. A key, with key space $2^k$ selects one of them. Although that's a huge number of keys, it is dwarfed by the amount of possible permutations.

Now it's not by definition impossible that the same permutation can be chosen by a key, but it is pretty unlikely. Such a key would be considered an equivalent key or - less specifically - a weak key. Many algorithms are designed to disallow weak keys. If it is possible to have the same permutation then it is likely to be present by design; you won't find one by chance alone.

Finding two keys that map a single specific plaintext to identical ciphertext or vice versa is much more likely. It depends on the block cipher details if this is anywhere near possible see this discussion on AES for more information.



For stream ciphers the question is somewhat more difficult. It probably suffices to say that the output of the stream cipher is determined by a key and an IV (sometimes fused into one, for instance in RC4). If the size of the key, IV and internal state of the stream cipher are large enough, you should not be able to find a large amount of repetition in the key streams generated for separate key / IV pairs.

It will be possible to find keys that generate small amounts of identical key streams. This is easy to see; about half of the keys should decrypt a single bit correctly. It may be impossible to determine if you'd have generated the correct plaintext though; you'd have to determine the two keys in advance.

The idea expressed above would also work for a block cipher in a streaming mode of operation such as CTR (counter mode).



These kind of questions have been asked, but they are tricky to find if you don't know the right terms - and probably even if you do.

Related is the question if algorithms contain back doors. In that case the attacker should be able to decrypt whatever the key value is though. 

Deniable encryption is another somewhat related term.