What had changed since the last time it worked and now? The only thing I could find was the keyboard and the mouse, both of which are now USB and were PS/2 before. Mmm... as a test, I pressed the suspend button and immediately afterwards unplugged both peripherals. Guess what? The machine got into sleep mode properly!
So, I opened the case and looked for the two jumpers in the motherboard (an Asus A7V8X-x) that tell it which power line to use for the USB ports: +5V or +5VSB. Changing them from the former to the latter fixed the problem and suspension works fine now.
Maybe it's time to try NetBSD-current to see if this feature also works in my machine...