When I was in my early 20s, my older brother (and only sibling) was overseas in the military and we had rare contact. We almost never exchanged emails or phone calls. One day our father died suddenly of a massive heart attack. I found out shortly after it happened, and proceeded to attempt to contact my brother, calling his house in Germany repeatedly across many hours (while attending to other family matters). While I was doing that, unbeknownst to me he was simultaneously sending me five or six emails out of the blue while at work, spanning various topics (he was entirely unaware our father had died). I didn't get the emails until after we spoke - I was busy trying to call him and with everything going on I didn't think to bother to check my email. He probably hadn't sent me an email in years. What are the odds he'd be trying to reach me at that particular moment, while I'm trying to call him about something so critically important to both of us.
In my early 20s, my mother died of small-cell lung cancer (30+ years of smoking), after about a 15 month fight with it. The disease progressed very rapidly in the later stages. In the last weeks, her drift toward death was essentially being managed, as there was nothing else that could be done (a hailmary round of intense radiation therapy dropped her off a final cliff, she died quickly after that). She was being cared for at home in the final two months, which is what she wanted. I would occasionally go visit and spend time with her, although she was out of it mentally in the final weeks, as her organs were gradually shutting down. I hadn't gone to visit her in a week or so. I unintentionally fell asleep the evening I planned to go visit, waking up late, a bit after 11pm; that was far too late to go visit unannounced, but I did it anyway, I felt compelled that night. So I drove to her house and I spent several hours with her, just the two of us. The TV or radio was always left on for her through the night (she couldn't communicate at this point), with an assumption that it might be comforting if she was still receiving input in some regard. I left at maybe 2am, turning off the TV as I left (I was supposed to leave it on, it was just an automatic action), and she died a few hours later that morning. Where did the strong compulsion to go see her that specific night come from? It was particularly odd, somewhere in my brain I knew I needed to regardless of how late it was, it couldn't wait until the next night.
I'm an atheistically and scientifically inclined type, non-religious, without a mystically-oriented mind, and these types of coincidences or oddities always throw me a bit regardless. If you live long enough you inevitably see many of them and they're always surreal. For the most part I chalk it up to the sheer volume of events (micro or otherwise) that occur across a lifetime, such that unlikely coincidences and strange events are bound to be produced.
The most common one seems to be where you're actively thinking about someone you haven't spoken to in a long while, and suddenly they call or text. It's always freaky and I don't think I've ever met someone it hasn't happened to.
I think our brains are subconsciously capable of rather remarkable calculations, estimations, extrapolations and timings that are still vastly underestimated. Roaming this planet with so many other people, the collisions of these capabilities produce remarkable coincidences as our daily lives unfold and cross paths. It makes sense, as we're all clones in the same human family (I don't mean that literally), we probably have the same impulses derived from a subconscious timer: both parties feeling a similar thing in the same rough time frame. It makes sense that we'd have similar bounds and reactions to reaching them, based on a shared relationship. And I think that mostly explains Twain's concept of mental telegraphy.