Hospital bills feels like a pretty ordinary compound to me - not like "good morning" or "ginger ale" where you can't just use what you know about the two words to figure out what the compound must mean.
Some cases are basically impossible "Crash blossoms" you don't stand any chance without knowing why we call them that
Some are middling difficult, "Home Secretary" requires that you know every meaning for the two words and then you happen to pick the correct obscure meaning, a "Secretary" could be in charge, and "Home" could mean the entire country as distinct from everywhere else.
But "Hospital bills" doesn't seem even marginally difficult
In most English speaking countries it's a far from common phrase (ie. it's very USA-centric).
Maybe you don't have "hospital bills". I don't have "landscaping bills", but I know exactly what they are.
As you've pointed out, the word "bills" clarifies what it is. I don't see why every combination needs to be in a dictionary. The list would be incredibly long, eg. "phone bills" or "power bills", etc.