(2017-03-20)
Problem
Can't run updates because /boot is full.
Solution if you are not hosed
Look in /boot, pick an old kernel version (make sure it isn't the one you are actively running by checking
uname -r) and delete it:
# apt-get remove linux-image-extra-4.4.0-{38,42,45,47}-generic
(If you are hooped, it will complain about unmet dependencies (which probably include a partially installed kernel that it can't complete because /boot is full) and you get to use a bigger stick, see the next section.
Once you've made space, fix up the un-met dependencies:
...and you should be good to go.
Solution if you are hosed
Use dpkg to forcibly uninstall kernels you are not using any more:
# dpkg --force-all -P linux-image-extra-4.4.0-{38,42,45,47}-generic
Now fix up the un-met dependencies:
...and you should be good to go… at least until it happens again.