I'm not sure what I did, but it seems like I can't see what anchor points I'm selecting with the direct selection tool. I can't round corners either, as the round corner dot doesn't appear. I can actually select them, as hitting the delete button does delete them. Has anyone encountered such an issue before?
Edit: I did some more tests, and it looks like it doesn't work within the particular document, for some reason. If I copy a path from that document and paste it into another one, it works.