I'm (also) having the issue of just getting a black screen when I try and play 400 Days (on OSX 10.8.4). After I press "Play", it asks me if it is OK to rewind or not, but regardless if I choose yes or no, I'm only presented with a blank screen.
I've tried verifying the cache files and renaming the save games folder so that steam resyncs the save game files, but this didn't work.
A wild guess would be that the prefs.prop file is somehow out of sync/corrupt. Anyhoo, I'm attaching the results of the support tool. Any ideas?