(Just helping Google out here.)
This seemingly weird issue of cold boot and every other boot thereafter failing (with successful boots in between) is mediated by Grub’s recordfail functionality: after a failed boot, the next boot is forced into linux_gfx_mode=text (see /boot/grub/grub.cfg), which then results in a successful boot.
The core issue is the failing boot, which in the case of Thinkpad X60 is probably LP #1724639 (although the report and comments are all over the place, making the report pretty useless).
To work around it, add GRUB_GFXPAYLOAD_LINUX=text to /etc/default/grub and run sudo update-grub.
This forces Grub to always boot in text mode, which is less pretty ofc, but at least it doesn’t hang.