Hi, just stumbled on this site and thought I'd jump in!
If you are standing still (compared to earth's surface) and watching two particles approach each other, you will see one moving right at almost c, and the other moving left at almost c.
Now imagine you are moving with one of the particles so that from your point of view, it is stationary. You see the other particle approaching you and measure its speed relative to you, and you find it is still less than c, rather than 2 x c as you'd expect.
How can these both be true at the same time? When you are moving with respect to another object, the space between you and the object 'contracts' (from the maths of special relativity). Therefore the distance you measure between the two particles is a lot shorter if you are travelling with one of the particles than if you are standing still in the lab.
Example: If, when travelling with one particle, the distance measured between the pair is 25% of that measured in the lab, their measured relative speed will be 25% of that you measure in the lab.
Edit: Alan, to answer your question about the observer, they will observe each particle moving (relative to them) at nearly c. The relative speed they will measure between the two particles to be almost 2c, and that's fine. You are not seeing anything moving at greater than c relative to you at any point in this process, and that's what counts!
Richard