None of those answers worked for me, running the game under XP. Sir Primalform's solution didn't help. I was already running the game as an administrator, and in any case I had ownership of the files. And, yes, I was being careful about which player name I used.
It took a few minutes of remembering back to my DOS days, but I finally realized what was wrong. I had installed the game to D:\Games\Theme Hospital.
The game's save feature can't follow a path that's not in 8.3 format.
I reinstalled to D:\Games\THosp and everything's working fine.