You can get away with testing over smaller distances as long as the precision is high enough. An answer to this can be found here: https://en.wikipedia.org/wiki/Tachyonic_antitelephone - the one-way example given there might not be that impressive (you can find a frame of reference where the superluminal movement leads to the arrival happening before the departure). But the two-way example shows a case where sending information into the past becomes possible if two people have access to such a device that can send information with superluminal speed. And that is true as long as the signalling speed is superluminal - that is why this is the speed of causality. Causes must happen before their effects. They are not ruled out by theory. Put together with the fact that there is no evidence that supports their existence and certainly no observed phenomenon that requires their existence in order to be explained, there is usually a very strong assumption that they don't exist.