It's unlikely, but possible, that someone could break ECDSA or SHA256, which would be bad for the Bitcoin community.
What would be the drawbacks of using two methods of encryption, one on top of the other, so that if either one is broken, Bitcoin remains unaffected?
I presume it would be far less likely for two distinct forms of encryption to be defeated at exactly the same time, than to have one broken.
Are there strong, well-tested encryption schemes that are based on mathematics sufficiently distinct from ECDSA?
Side question: Do there exist viable forms of cryptography which are thought not to be defeatable by a quantum computer? I've heard of "Unbalanced Oil and Vinegar" which is supposed to be quantum-computer-resistant but not thoroughly tested.
Keep in mind a "break" never shatters an encryption or hashing method. It always happens in small incremental improvements on brute force techniques.
That said, doubling up on the hash should be simple and would likely improve security. I believe layering ECDSA with another signing/verification method would be a little more challenging, but I don't have a strong enough crypto background to delve into how that might work.