What is the output of the 'df -h' command (to be entered in a terminal)?
Do you remove the older kernels? You normally only run the newest installed kernel and had better keep an older one you are sure you can boot. But there is no need for more than that. Yet, when a newer version of the kernel is installed, the older ones remains and occupy space in /boot. You can remove the useless "linux-*" packages from the "Synaptic package manager" (in the "System settings").
If /boot is not on a separate partition, then the whole root partition probably is full. You can analyze where the space goes with the "Disk usage analyzer" (in the "System settings" too). If you have the system for years but never clean the packages (with 'sudo apt-get autoclean' or, more radical, 'sudo apt-get clean') then, that might be a reason.
It may also be a log file that grows out of hand. Anyway, you had better take a look at what the "Disk usage analyzer" says.
