The line charges must have some mass. Because of this it is impossible to accelerate them to the speed of light.
Increasing their speed also increases their inertia. Initially almost all of the work done increases their speed and almost none increases their inertia. As their speed approaches the speed of light, almost all of the work done on them increases their inertia and there is little increase in speed.
So their speed will always be less than the speed of light, although you can get as close to $c$ as you wish.