File system bafflement

[R Kimber]

I'm at a loss to understand what's going on on my system.  I keep getting
messages that the root filesystem is full.  Since the entire system is
contained in / I'm not clear what this means.  Does it just refer to the
top directory level, or does it include subdirectories apart from those
with their own partitions, or does it refer to the whole lot?

The disk usage analyser also claims that /media contains 2.4 TB and is is
"96% full", but /media surely is only the mountpoint for other disks. There
are, inter alia, two 2TB disks mounted there. The disk usage analyzer says
they are 49.9% used, while df says they are 66% used.

The system is also a bit slow, though there's plenty of swap space available.

Yesterday I cleared the apt cache, and the wastebin, and removed some
kernels, and reduced / usage to about 84%, but today it's full again. It's
hard to see where the problem lies since I've not done anything that
involves the / partition that I'm aware of.  Though I noticed that an
external USB disk had disk errors, which e2fsck corrected (Free blocks
count wrong - maybe the disk is failing) - does this have anything to do
with it?  

All the following have their own separate partitions:
/usr, /var, /tmp, /home, /usr/local
and df reports that they are all less than 50% full

The size of the root partition is 4GB, and the top level directories that
do not have their own partitions amount to just over 1GB

My system has 4GB of memory and is running 14.04 (3.13.0-35-generic) that
was upgraded from 12.04LTS.  I'm using the mate desktop.

I'd appreciate suggestions as to what the problem might be.

- Richard.
-- 
Richard Kimber