From the start, I will state my opinion that although there have been many tests, some by very respectable institutes on the subject of telepathy, I do not believe that on the balance of evidence, telepathy can be placed under the subject of science, but only as a part of pseudoscience. However, as usual, I will present some of the evidence for and against and you can come to your own conclusion.
Telepathy is supposed to be the transmission of thoughts from one person to another. It got its name in 1862 when the founder of the Society for Psychical Research, Frederic Myers. In the late nineteenth century, a number of people claimed they could identify objects and marked cards whilst being blindfolded. They did not claim to be telepaths but they did insist on holding the wrist of someone who knew the answer. It was this, they said that allowed them to guess the hidden object as slight ideo-motor movements gave them clues.
The twentieth century saw many various types of scientific research into telepathy. These were in parallel with many show people pretending they could read minds to paying audiences. The result of all this was that telepathy was still not considered to have any scientific foundation.
However, this did not stop some countries such as the USA and Russia from experimenting with telepathy and remote viewing, which is observing a location many miles from the observer; for example, spying on Russia from the USA. A great deal of time and money was spent on these experiments during the Cold War. However, this was a time when the superpowers were desperate to try anything that might give them an advantage. But, it will come as no surprise that nothing positive came out of it. At least, that’s what military intelligence has told us.
Of course, there have been sworn statements of people claiming they had received a thought transmission warning them of some disaster which then occurred. My view of these few cases is that they are the exceptions that prove the rule. In other words, I put it down to coincidence.