You can still make an appointment at the Ausländerbehörde (foreigner's office) a couple months in advance, and then you just have to show up on time. And as recently as a year ago, the last time I needed something same-day, showing up two hours before they opened (6 a.m.) was still enough to be at the front of the line. There were at that time not a fixed number of tickets for the day. Historically (meaning over the course of the decade I've lived in Berlin), if you needed an appointment same day, at any time, you'd usually wait less than two hours, no matter when you showed up. Much of this has worsened recently because of the refugee crisis, but it's a temporary overload of the system, not a systematic failure.
