Ok, figured it out myself! I had used add_filter to customize the password form in order to add css classes. Everything in add_filter was either old or simply wrong. I found it in a tutorial, but this has been a lesson to me. Never trust code found on the web unless it from the Codex or an absolute expert.
EDIT: Just found the same piece of code in the Codex as well *facepalm. It says “THIS NO LONGER WORKS” above it, but then why is it still in the codex? To anyone having this problem as well: in case you have used the code from here: Using Password Protection: Password Form Text you’ll get problems in newer versions of WordPress.