I installed a newer kernel from Debian Backports, but during the reboot I noticed an error message flash by- far too quickly to read.
I was sure I had found a way to read such messages again after the computer finished booting, but just couldn't recall it.
After spending a couple of hours on Google trying to rediscover the secret, I found it.
I've put it in red and highlighted it so I don't forget it again.
Running dmesg showed me the error message I'd missed, highlighted in red.
firmware: failed to load radeon/mullinsMy new kernel couldn't use the non-free driver firmware I had installed- installing firmware from Debian Backports fixed the problem.