One of the requirements for a hash to be cryptographically secure is that all possible values are equally likely. So yes, it is possible (in fact trivial) to construct a hash of the sort you describe, but such a hash would not be cryptographically secure to begin with.
It's easier to use and to reason about a uniform hash, but you can design a secure system with a non-uniform hash.
I'd be willing to bet that an altered version of SHA3-256 that replaces four bits in the middle with length%16 is better for most purposes than SHA256, despite being non-uniform.
