You probably really ought to ask somewhere that deals with statistics.
But the brief answer is as follows: yes, anecdotes can be informative. They can't be tested since there's only one of them, but they can still be highly suggestive. However, this is only true in Sherlock Holmes style: when you have eliminated the impossible, whatever remains, however improbable, must be the truth. This also works if "impossible" just means "way, way less probable than 'however improbable'".
The intuition is pretty clear. Swallowing a battery-sized model rocket engine ought not make someone explode. It might make them sick (it's probably poisonous) or hurt them, but not explode. But people really really don't go around exploding much.
Suppose you, one time, see someone gulp down a model rocket engine and then explode. Heck--you don't even need to see it, as long as you know it happened by some reliable source (i.e. it's much more likely that it happened, weird as it is, than that your source is wrong).
Now, you think: well, people don't go around exploding under normal circumstances; this would be incredibly incredibly weird that just in those few seconds after swallowing a model rocket engine he'd explode for some other bizarre reason. So, maybe model rocket engines actually can make someone explode.
You can quantify this with Bayesian statistics (and find that it is valid, at least with reasonable distributions of priors).
The problem with anecdotes is that people habitually underestimate the chance that the account is wrong, or fail to realize that the world has seven billion people in it and all sorts of stuff happens by chance and we select out the weird-seeming things to pay attention to. So if you want a rule of thumb: ignore anecdotes. But if you want to be statistically accurate: yes, they contain information (and can possibly justify you changing your model of what is likely by quite a bit*).
*(Addendum - the clearest case where an anecdote has huge power is a single instance of something happening that is said to be completely impossible. For example, if "all ravens are black", and you see a white raven, that's pretty good evidence against "all ravens are black". Of course it might not actually be a raven so without a lot of investigation it's not conclusive, but single examples can go a long way towards falsifying statements.)