What are some examples (from the public sphere or not), of overthinking something?

In some cases, all possible information from any angle is important, such as measurements of height, length, velocity, and probability scales for success in jumping the grand canyon with a motorcycle.

Overthinking normally is attributed to a more psychological angle.
If a young man, for instance, wants to presents a lady with flowers, and feels nervous. He might overthink the situation by imagining all possible outcomes, including her father might be a veteran suffering ptsd and would shoot him.
That would be dredging up a scenario with almost no possibility of being fact.
Overthinking situations tends to happen most often when someone hopes to talk themselves out of their own idea because of fear.

People in some relationships – mostly but not exclusively women – that try to anticipate every possible thing that could ever go wrong. And then try to avoid it or nip it in the bud. Instead of just enjoying life and experiencing good things, they overanalyze every word that the partner says in order to suss out the worst possible meaning.

Thanks @Patty_Melt and @elbanditoroso

Now I’m trying to think if some well known (maybe not so well known, just public) event where things turned out well as a result of someone overthinking it.

While I am certainly not a proponent of anticipating others’ moves, I have been accused—and recently, too—of overthinking. I think people don’t think enough, stopping short of seeing the whole picture.
In my most recent incident, it was a Bible question. What can I say? They were wrong, and I was right. I laugh, but it is true: they were wrong and I was right.
When I run across this, I say it once, they don’t get it, I say it one more time in a simpler way, they still don’t understand, so I go silent.