"Regular medicine has antidotes: alternative medicine has anecdotes." - Frank Coffman
One of the scientific rules is that anecdotal evidence doesn't cut it. Suppose I tell you that I thought of a long-lost friend. Just then the phone rang, and it was my old friend. That's a nice anecdote. So why doesn't it prove anything about precognition?
If this story proved anything, it would be proving that the unlikely had happened. But there's no way to tell if the event was in fact unlikely. Perhaps I think about the friend a lot. Perhaps I've thought of people a zillion times, and only one of them phoned. And there's the lottery-ticket thing. The person with the winning ticket sees that as very unlikely. But someone had to win. So, how many other people thought of a long-lost friend, and got no call? I don't know the answer. And until I know the answer, the anecdote doesn't prove anything.
A large collection of anecdotes isn't necessarily any better than a small collection. There are several famous cases where large groups of people all swore that some new remedy was a wonder cure. (For example, Mesmer's "animal magnetism".) The main value of the collection is that their number might inspire someone to look into the matter properly.
Anecdotes about health are often useless. This is because some number of patients will get well anyway, no matter what treatment they receive. If a cancer goes into remission, it is very human to think that some recent change caused a cure.
Anecdotes are also entirely too susceptible to being unverifiable. If someone tells a story, and you press him for details, it is entirely likely that he doesn't have them. He can't remember, it was long ago. Or, no doctor ever actually verified the disease, much less its cure. Or, it turns out it didn't actually happen to him: it happened to his sister. If you force him to contact his sister, she remembers it differently. Or, worse, it turns out it happened to a friend of the sister. We are now chasing down a FOAF (Friend Of A Friend) story. FOAF stories are usually rumors or "urban legends".
In short: anecdotes, bleah.