There is a story that goes the rounds in various versions, but the one I like best is about the area I live in. The people involved could have known each other, but I have no direct evidence that they did. It goes like this:

There was a pub down by the river in Kingston where members of the acting profession used to gather. A member of the public (hereinafter referred to as "the motp") went up to a table where a number of people were enjoying a quiet drink together and addressed one of them:
"You're him, aren't you? You're in that thing we watch, aren't you? Now, what's your name? No, don't tell me. I'm sure I know it."
After an embarrassingly lengthy period of fruitless dithering, the motp gave up and asked the familiar actor to identify himself.
"Well, actually, my name is Allan Cuthbertson." The Anglo-Australian actor Allan Cuthbertson did live in Surbiton, which is next door to Kingston, so he may well have used that pub.
However, the motp was not convinced.
"No, you're not Allan Cuthbertson. You don't look anything like him."
"Well, I can assure you that that is my name."
Now, I suspect the motp was confusing Allan Cuthbertson with the Scottish actor Iain Cuthbertson, who was also familiar to television audiences at that time. But the motp persisted:
"If you're Allan Cuthbertson, then I'm James Bond."
Whereupon one of the other people who had been sitting quietly, patiently enduring this intrusion, looked up and said,
"No, you're not. I am."
The motp had failed to recognize Sean Connery.

I cannot attest to the accuracy of this story, but I suspect it is the sort of thing that happens to celebrities. Incidentally, I once saw Allan Cuthbertson walking along in Surbiton near where Waitrose now is. I made a point of not appearing to notice him.