The easiest way might be to boot the system that's having problems and disable the splash screen. That will then show the boot messages during bootup.
To show the boot messages, start up and when grub appears press 'e' to edit the grub entry. Press the down arrow until you get to the linux kernel line. Press CTRL-E to go to the end of the line. Remove
quiet splash from the line. You might also need to delete any line that begins with
gfxmode if it still doesn't work. I had to remove the gfxmode line to be able to see the boot messages on Xubuntu. Now press F10 to boot. The boot messages should appear.
Something else you can try is to boot from your rescue disk, mount the filesystems of the computer, then look at
/var/log/boot. The last one may not exist or be empty if the computer is not configured to run bootlogd at startup. You can edit
/etc/default/bootlogd and make sure that
BOOTLOGD_ENABLE=Yes is set, then reboot, to get the boot output into
Between all of those options you should be able to determine where the system is getting stuck at bootup.