If you look at the literature of the past century or so, there have been many modern people whose worldview was mostly secular and scientific, and skeptical of religion, who believed in psychic phenomena. To such an extent that science fiction uses psychics, telepaths, and so on in a quite straight faced manner. However, the evidence has never been found to substantiate their belief. Many prizes have been put up for evidence of the paranormal, and they have not been claimed.

So what's going on? What seems obvious to me now is that the belief in paranormal phenomena is something that we tend to acquire for reasons other than evidence. For example, we often get a hunch based on patterns we see -- patterns can in some situations be good evidence, but also can be misleading. Like a gambler who thinks they are having a "lucky streak" because they have won a few times in a row (when in reality their chance of winning an additional time is not changed relative to if they have lost the last few times).