For anyone trying to find an answer to this, here is what you can do You are going to be replacing the kernel values and versions with your own system. Don't attempt to use mine, exactly...
First, determine your current kernel version.
uname -r
That will be the kernel you want to make sure you don't delete.
List the kernel packages installed.
sudo dpkg --get-selections | grep linux-image
linux-image-3.13.0-106-generic install
linux-image-3.13.0-117-generic install
linux-image-3.13.0-45-generic deinstall
linux-image-3.13.0-61-generic install
linux-image-3.13.0-91-generic install
linux-image-3.13.0-98-generic install
linux-image-3.19.0-64-generic install
linux-image-3.19.0-71-generic install
linux-image-3.19.0-78-generic install
linux-image-extra-3.13.0-106-generic install
linux-image-extra-3.13.0-117-generic install
linux-image-extra-3.13.0-45-generic deinstall
linux-image-extra-3.13.0-61-generic install
linux-image-extra-3.13.0-91-generic install
linux-image-extra-3.13.0-98-generic install
linux-image-extra-3.19.0-64-generic install
linux-image-extra-3.19.0-71-generic install
linux-image-extra-3.19.0-78-generic install
linux-image-extra-3.19.0-80-generic install
linux-image-generic install
linux-image-generic-lts-vivid install
In my case, apt was trying to install 3.19.0-80, but could not, because /boot was full.
Get a list of all the kernel image files in the /boot directory.
christopher@sbs-server:/boot$ ls
abi-3.13.0-106-generic initrd.img-3.19.0-64-generic
abi-3.13.0-117-generic initrd.img-3.19.0-71-generic
abi-3.13.0-61-generic initrd.img-3.19.0-78-generic
abi-3.13.0-91-generic lost+found
abi-3.13.0-98-generic System.map-3.13.0-106-generic
abi-3.19.0-64-generic System.map-3.13.0-117-generic
abi-3.19.0-71-generic System.map-3.13.0-61-generic
abi-3.19.0-78-generic System.map-3.13.0-91-generic
config-3.13.0-106-generic System.map-3.13.0-98-generic
config-3.13.0-117-generic System.map-3.19.0-64-generic
config-3.13.0-61-generic System.map-3.19.0-71-generic
config-3.13.0-91-generic System.map-3.19.0-78-generic
config-3.13.0-98-generic vmlinuz-3.13.0-106-generic
config-3.19.0-64-generic vmlinuz-3.13.0-117-generic
config-3.19.0-71-generic vmlinuz-3.13.0-61-generic
config-3.19.0-78-generic vmlinuz-3.13.0-91-generic
grub vmlinuz-3.13.0-98-generic
initrd.img-3.13.0-106-generic vmlinuz-3.19.0-64-generic
initrd.img-3.13.0-61-generic vmlinuz-3.19.0-71-generic
initrd.img-3.13.0-91-generic vmlinuz-3.19.0-78-generic
initrd.img-3.13.0-98-generic
Create a folder christopher@sbs-server:/boot$ mkdir ~/old-kernel-images
Move all the old kernels to the above folder. christopher@sbs-server:/boot$ sudo mv ./*-3.13.0-* ~/old-kernel-images
Now, you should have space in /boot, but apt will complain about missing files, when you install the latest kernel version. We'll handle this, later.
update-initramfs: Generating /boot/initrd.img-3.13.0-106-generic
grep: /boot/config-3.13.0-106-generic: No such file or directory
update-initramfs: Generating /boot/initrd.img-3.13.0-98-generic
grep: /boot/config-3.13.0-98-generic: No such file or directory
update-initramfs: Generating /boot/initrd.img-3.13.0-91-generic
grep: /boot/config-3.13.0-91-generic: No such file or directory
update-initramfs: Generating /boot/initrd.img-3.13.0-61-generic
grep: /boot/config-3.13.0-61-generic: No such file or directory
Once it is finished, purge all the missing kernel packages. This, also, will give errors, but it is necessary to tell apt they are no longer installed.
christopher@sbs-server:/boot$ sudo apt-get purge linux-image-3.13.0-91-generic linux-image-3.13.0-61-generic linux-image-3.13.0-45-generic linux-image-3.13.0-117-generic linux-image-3.13.0-106-generic
Once everything is purged out, you should no longer have any issues.
christopher@sbs-server:/boot$ dpkg --get-selections | grep "linux-image"linux-image-3.19.0-64-generic install
linux-image-3.19.0-71-generic install
linux-image-3.19.0-78-generic install
linux-image-3.19.0-80-generic install
linux-image-extra-3.13.0-45-generic deinstall
linux-image-extra-3.19.0-64-generic install
linux-image-extra-3.19.0-71-generic install
linux-image-extra-3.19.0-78-generic install
linux-image-extra-3.19.0-80-generic install
linux-image-generic-lts-vivid install
Now install your latest kernel. There should be no errors.
christopher@sbs-server:/boot$ sudo apt-get install --reinstall linux-image-3.19.0-80-generic
Clean up any straggling packages, using autoremove. There should be no errors.
christopher@sbs-server:/boot$ sudo apt-get autoremove