Linux kernel images are stored on a separate partition mounted under /boot. This partition can fill up, at which point you can no longer install any software updates.
Ubuntu (and possibly Debian and Mint) has a command called purge-old-kernels that helps to prevent you from ever getting in that situation. Similarly, RHEL/CentOS/Fedora have a command called package-cleanup.
However, if your /boot partition is already full, purge-old-kernels won’t work. You will need to run something like the following:
dpkg --list 'linux-image*' | cut -d' ' -f3 | grep linux-image | grep -v "$(uname -r)" | grep "[0-9]" | xargs dpkg -r --force-depends
apt-get -fy install