For instance: two protons are brought to collision at CERN'S LHC. An observer located at the center of the detector measures a reduction of the distance between both particles until the particles collide. From the reference frame of each particle, the relativistic formula for the addition of velocities yields a relative velocity smaller c. The observer at rest 'sees' both particles symmetrically approaching. So what is the speed of that reduction of distance between the protons, as seen by the observer?
The speed in question is just under the double the speed of light. This however does not create any relativity contradiction. In relativity, the local speed of one object relative to another cannot exceed the speed of light. In your thought experiment, every relative speed is under the speed of light. The speed you are describing is not a relative speed, but two independent relative speeds. One is the speed of the first proton relative to you; the other is the speed of the second proton relative to you. You can combine these speeds the way you like and the way you are describing they combine to near double the speed of light. However, there is no such a concept in relativity, as the speed of one object relative to the other in the frame of the third (e.g. observer). "Relative" means one relative to the other in its frame, not in the frame of the third party. If you calculate the speed of one proton relative to the other proton, the result would be under the speed of light.