Are night trains (particularly sleepers) generally fully booked? I'm wondering if I need to book in advance, or can I do it the day before?
One day might be pushing your luck. We have tried to book sleepers at least three days before we are leaving.
A few years ago, I tried to help a woman who was trying to book the night train from Munich to Venice. It was a month in advance and all the sleeping accommodations were booked. There was nothing left but seats. Yes, it can happen. Book as soon as you can commit. Also, if you are trying to get discounted tickets, the lowest priced ones go quickly.
Absolutely you need to book ahead. It depends on the route and the time of year of course, but they often fill up. Book at least a few weeks in advance if possible.