It's probably because you have too many kernel packages installed (you can only run one at a time).
Check which kernel packages are installed:
dpkg -l 'linux-image*' | grep '^ii'
Check what your current kernel is:
uname -r
And just remove the older versions. For instance on my machine my /boot directory contents looks like this:
% ls -al /boot
drwxr-xr-x 4 root root 1024 May 2 13:00 .
drwxr-xr-x 27 root root 4096 Apr 18 01:09 ..
-rw------- 1 root root 3018381 Mar 11 22:39 System.map-3.5.0-26-generic
-rw------- 1 root root 3020028 Mar 26 19:54 System.map-3.5.0-27-generic
-rw------- 1 root root 3020830 Apr 24 23:04 System.map-3.5.0-28-generic
-rw-r--r-- 1 root root 850088 Mar 11 22:39 abi-3.5.0-26-generic
-rw-r--r-- 1 root root 852365 Mar 26 19:54 abi-3.5.0-27-generic
-rw-r--r-- 1 root root 852490 Apr 24 23:04 abi-3.5.0-28-generic
-rw-r--r-- 1 root root 147953 Mar 11 22:39 config-3.5.0-26-generic
-rw-r--r-- 1 root root 148105 Mar 26 19:54 config-3.5.0-27-generic
-rw-r--r-- 1 root root 148105 Apr 24 23:04 config-3.5.0-28-generic
drwxr-xr-x 3 root root 5120 May 2 13:01 grub
-rw------- 1 root root 23540031 Mar 28 18:31 initrd.img-3.5.0-26-generic
-rw------- 1 root root 23574269 Apr 9 08:26 initrd.img-3.5.0-27-generic
-rw------- 1 root root 23578154 May 2 13:00 initrd.img-3.5.0-28-generic
drwxr-xr-x 2 root root 12288 Oct 5 2012 lost+found
-rw-r--r-- 1 root root 176764 Nov 27 2011 memtest86+.bin
-rw-r--r-- 1 root root 178944 Nov 27 2011 memtest86+_multiboot.bin
-rw------- 1 root root 5180064 Mar 11 22:39 vmlinuz-3.5.0-26-generic
-rw------- 1 root root 5180864 Mar 26 19:54 vmlinuz-3.5.0-27-generic
-rw------- 1 root root 5183296 Apr 24 23:04 vmlinuz-3.5.0-28-generic
I have the following kernel image packages installed:
% dpkg -l 'linux-image*' | grep '^ii'
ii linux-image-3.5.0-26-generic 3.5.0-26.42~precise1 Linux kernel image for version 3.5.0 on 64 bit x86 SMP
ii linux-image-3.5.0-27-generic 3.5.0-27.46~precise1 Linux kernel image for version 3.5.0 on 64 bit x86 SMP
ii linux-image-3.5.0-28-generic 3.5.0-28.48~precise1 Linux kernel image for version 3.5.0 on 64 bit x86 SMP
ii linux-image-generic-lts-quantal 3.5.0.28.35 Generic Linux kernel image
And my currently running kernel is:
% uname -r
3.5.0-27-generic
Thus, in my case I could free up space in /boot by removing:
linux-image-3.5.0-26-generic
linux-image-3.5.0-28-generic
by running apt-get remove <package_name>