I suspect that it's not possible, but it's not a question that armchair reasoning will let us answer. There certainly are magician's tricks that simulate telepathy. There are also experiments that are suggestive of something more, though they hardly amount to full-blown proof. The most interesting experiments I know of have to do with the so-called "Ganzfeld" effect. Subjects ("receivers") are put into a mild state of sensory deprivation and talk about what is going through their minds while a "sender," in another location, looks at one member of a set of four images. Later, the "receiver" is shown the four images and asked to pick the one the "sender" had been looking at. On some readings of the evidence, receivers are able to pick the correct target at a rate significantly above chance. The Wikipedia account is a good summary of the experiments and the controversy. As you'll see, the results are hardly overwhelming, and there's plenty of room to argue, but there's at least some room to take this as serious evidence of a weak telepathic effect.

My point is not to endorse that conclusion. It's just to stress that there is an empirical question here. Some philosophers have argued that there could never be evidence good enough to properly convince us that ESP is real. They make their case by adapting David Hume's arguments against miracles. A bit too briefly, the idea is that we have overwhelming evidence from other parts of science that nature doesn't work as it would need to for ESP to be real. We also have plenty of evidence that people lie, cheat, get caught up in ideas and inadvertently jigger the evidence, and so on. The verdict is meant to be that it's much more likely that there's something wrong with experiments like the Ganzfeld experiments than that something as scientifically puzzling as telepathy is real.

That way of looking at things carries some weight, but strikes me as inconclusive. After all, there are certain real-world physical effects that would have seemed too weird to be taken seriously not so many years before they were established. There's plenty of reason to be skeptical about telepathy, but we can't show by mere philosophical argument that the world is as tidy and well-behaved as we might like.

But we can add: there's a view about cause and effect that goes back to Hume and that bears on the larger issues. Very roughly, the idea is that cause and effect, laws of nature, and so on are ultimately nothing more than regularities in the patchwork of events. There's no necessary connection between cause and effect; it's simply that nature happens to display certain patterns. There's no deep reason for whatever patterns happen to hold; they just are. If that's right, however, ESP, if it were real, wouldn't be "supernatural" or contrary to materialism or scientifically forbidden. It would just be a surprising higher-level pattern in the cosmic flux.

In short, then, while there is some evidence on behalf of telepathy, it's very far from making a strong case. And even if the evidence pans out, accepting it might be less disruptive to our usual world-view than one might have thought.