No, it is not possible.
One private key has one public key. So each address has only one private key.
Ok. I stand corrected. I previously thought that you could have lots of private keys hashing to a single bitcoin address, but that collisions would still be so rare as to render that non-problematic.
The thing that I find interesting is that a private key is enormous compared to the size of a bitcoin address, leading me to believe that it would be possible to hash multiple private keys to the same bitcoin address.
You are correct. There are potentially nearly 2
256 public keys, and there can't be more than 2
160 bitcoin addresses. Therefore, there are likely to be multiple public keys (and therefore multiple private keys) that all result in the same bitcoin address. Fortunately, there is no known way to find these additional keys that result in the same address.