One such story is for example that of Niels Bohr and the horseshoe. The version on this website goes like this:
"An American scientist once visited the offices of the great Nobel prize winning physicist, Niels Bohr, in Copenhagen. He was amazed to find that over Bohr's desk was a horseshoe, securely nailed to the wall, with the open end up in the approved manner (so it would catch the good luck and not let it spill out). The American said with a nervous laugh,
"Surely you don't believe the horseshoe will bring you good luck, do you, Professor Bohr? After all, as a scientist --" Bohr chuckled.
"I believe no such thing, my good friend. Not at all. I am scarcely likely to believe in such foolish nonsense. However, I am told that a horseshoe will bring you good luck whether you believe in it or not."
In some other versions that you can find online it's a student who asks the question, in yet some other versions the horseshoe is not above the desk but above the door to Bohr's cottage. The above version is particularly interesting for the amount of irrelevant details that somebody or maybe several people have added. Wikipedia lists the quote as disputed.
To find the origin of that story it is useful if you speak German, since it goes back to Werner Heisenberg's book "Der Teil und das Ganze" (The part and the whole). Most of the book is a recollection of conversations Heisenberg had with Niels Bohr and Wolfgang Pauli, among others. Heisenberg wrote down these conversations long after they had taken place, so one should not expect the exchange to have been word by word exactly as he reported. Heisenberg finishes Chapter 8 on "Atomphysik und pragmatische Denkweise" (atomic physics and pragmatism) with an anecdote that Niels Bohr told:
Niels schloß das Gespräch ab mit einer jener Geschichten, die er bei solchen Gelegenheiten gern erzählte: "In der Nähe unseres Ferienhauses in Tisvilde wohnt ein Mann, der hat über der Eingangstür seines Hauses ein Hufeisen angebracht, das nach einem alten Volksglauben Glück bringen soll. Als ein Bekannter ihn fragte: "Aber bist du denn so abergläubisch? Glaubst du wirklich, dass das Hufeisen dir Glück bringt?", antwortete er: "Natürlich nicht; aber man sagt doch, daß es auch dann hilft, wenn man nicht daran glaubt.""
Niels finished with one of these stories he liked to tell on such occasions: "Near by our vacation house in Tisvilde lives a man who has a horseshoe above his door, after the old superstition that it brings luck. When a friend asked him "Are you superstitious? Do you really believe the horseshoe brings luck?" He replied "Of course not; but they say it also helps if you don't believe it."
~Werner Heisenberg, Der Teil und das Ganze, 1973, p. 112/13
So if you plan on winning a Nobel prize, be careful with the anecdotes you tell. Later generations might unashamedly turn the narrator into a subject and the details may be buried in translation.