The conjunction fallacy is the way a story can sound more plausible if you add more details to it, even though in reality every extra detail makes the whole story strictly less probable.
For example:
The statement "Bill is an accountant who plays jazz for a hobby." is less probable than the statement "Bill plays jazz for a hobby."
Intuitively, you might expect that Bill is more likely to be an account who plays jazz, than he is to play jazz.
This intuition is wrong, because the conjunction rule of probability theory states that, for all X and Y, P(X&Y) <= P(Y).