Consider two electrons moving parallel to each other in the same direction with same constant velocity. Will they experience any force due to either of them?
Answer
Yes, they will feel both electric and magnetic force. If you apply a Lorentz boost and get into the frame where they are at rest, they will simply feel the electric field of each other.
However if we stay in the lab frame we will notice that the force they experience goes down with $1/\beta \gamma^2$ (relativistic factors), approaching zero as they approach the speed of light. You can picture this also noticing that the time is flowing slower in their fast moving frame of reference and so you kinda see their radial movement in slow-motion.
This is a key point in accelerator physics where you want to get the beam of particles travel as fast as possible in the minimum amount of space thus reducing the radial defocusing effect due the so called "space charge".
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