It's always possible!  But for sufficiently "scrambling" functions like SHA256 (or cellular automata like Wolfram's rule 110, etc.), my guess would be that even the hidden structure, once found, would at most let you shave a small factor off from brute-force search.  If you told me that you could invert one specific such function in polynomial time, I'd be shocked but would get over it.  But if it turned out to be true again and again for many such functions, that's when I'd start wondering if one-way functions didn't exist after all, or indeed P=NP.

ScottAaronson

For things like SHA-256, we basically assume they behave like random functions with no useful exploitable structure. Do you think there’s any realistic chance that hidden structure could make them easier than we expect, a bit like how many-body systems can look intractable until some emergent structure appears? I’m not specifically thinking about quantum algorithms here.

I'm Scott Aaronson, quantum computing theorist at UT Austin!  AMA

From @south_korea_ln 3: 

> For things like SHA-256, we basically assume they behave like random functions with no useful exploitable structure. Do you think there’s any realistic chance that hidden structure could make them easier than we expect, a bit like how many-body systems can look intractable until some emergent structure appears? I’m not specifically thinking about quantum algorithms here.