I had the same issue with /boot being full.
I did some reading on the issue, found several forums dealing with this based on ubuntu, but some on raspberry/raspbian.
Some said you could delete/move "kernel7.img" for example, as the kernel was only used upon boot, and as you can't find kernel version on raspberry (by typing uname -a for example) which you can on ubuntu, that would mean that it was kernel.img that are being used.
I'm not sure if this is correct but that's what the forums said.
So I did move the kernel7.img to /home/USER and could then run the upgrade.
But after reboot it never went up again...
It turned out the SD card got corrupted...
So I'm not sure it's the right way to free up space in /boot this way?
(I had backup so no worries :) )
After the restore, /boot was not full?
But there seems to be some issue with the update process from 1.5 to 1.51, see
https://forum.category5.tv/thread-575.html