This link shows what kind of dialog there is:
http://elementaryos.stackexchange.com/q/1634
The issue is that the dialog tells to empty trash and use 'sudo apt-get clean'
which may help only, if /boot is on same partition than some of the files in
trash, the /var/cache/apt/archives/ or /var/cache
In 14.04 there is a dialog that appears informing you should free space
on /boot, if there is too little free space, when using Software
Updater. Not sure, if it works on later releases.
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** Summary changed:
- Old kernels filling up /boot, causing failed updates
+ Software Updater does not guide user to remove old kernels when /boot is
about to get full, causing failed updates
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