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.