If you have run out of space in /boot and running the command below doesn't appear to be helping:
sudo apt-get autoremove
Then you may need to manually remove some of the old kernels.
Find out the kernel you are running right now:
Make note of this as you must not remove this kernel in the steps below whilst you are using it.
List the images in /boot with
ls /boot. You should get output similar to:
abi-3.19.0-33-generic memtest86+.bin config-3.19.0-33-generic memtest86+.elf grub memtest86+_multiboot.bin initrd.img-3.19.0-33-generic System.map-3.19.0-33-generic lost+found vmlinuz-3.19.0-33-generic
Append the numbers of the kernels you don't want to the command below:
sudo apt-get remove linux-image-
E.g. if you see
vmlinuz-3.19.0-33-generic and you want to remove it you would execute:
sudo apt-get remove linux-image-3.19.0-33-generic
Reinstall Latest Kernel
In case your last update failed because /boot was filled, you will want to reinstall the latest kernel and update grub.
Run the command below to get a list of possible kernels:
sudo apt-get install linux-image
Now reinstall the one you want, for example:
sudo apt-get install --reinstall \ linux-image-3.19.0-33-generic
lastly update grub