I was in the middle of level 4 when patients stopped showing up to my hospital. I got an emergency at one point and a couple of VIPs, but otherwise it's just my staff wondering around aimlessly. I tried restarting and clicking the thing to open the hospital and still nobody showed. Please tell me I haven't encountered some sort of game-breaking bug.

I've never had such a situation, and i finished the game. I'm wiling to bet it is indeed a pretty bad bug. Sorry about that.

Same thing happened to me. I've got something like 7000+ visitors and all of a sudden visitors stopped coming in. Let me know if you found a way to resolve it

You might want to check your reception desk queue, if you (accidentally) set that to 0 you wont get any visitors. It also needs a receptionist standing at it ;)