I used rpi-update to update Raspbian, then restarted it. It got through the boot screen, but then just showed a black screen with a flashing cursor. For some reason, the monitor's backlight was pulsing every second or so as well. The LEDs were all on on the board except for the OK LED.
I reinstalled raspbian with Win32DiskImager, but the same thing is still happening.
The last messages shown before the screen goes blank are:
Starting dphys-swapfile swapfile setup ...
want /var/swap=100MByte, checking existing: keeping it
[ ok ] Starting Light Display Manager: lightdm
What's wrong with it?