"Extraordinary claims require extraordinary evidence" -- and I wouldn't count these small studies as extraordinary. Unfortunatley, publishing these kind of claims prematurely help the more gullible among us to fall for ridiculous claims from psychics and others who would take advantage of them. (The authors of "The Secret", I'm looking at you.) Commenters on NPR's website (not exactly the dumbest audience online) hav…
True. But on the other hand, publishing ridiculous claims and incorrect results is a necessary part of science.
When we publish only results we know to be correct, because they agree with mainstream beliefs, we introduce a bias into the scientific process. In reality, if you publish 20 experiments with p=0.05 [1], 1 of them should be incorrect. If less than 1 in 20 of your papers isn't wrong (assuming p=0.05 is the gold standard), you are not doing science.
You can see a perfect illustration of this when people tried to reproduce Millikan's oil drop experiment. I'll quote Feynman: Millikan measured the charge on an electron...got an answer which we now know not to be quite right...It's interesting to look at the history of measurements of the charge of an electron, after Millikan. If you plot them as a function of time, you find that one is a little bit bigger than Millikan's, and the next one's a little bit bigger than that, and the next one's a little bit bigger than that, until finally they settle down to a number which is higher.
Why didn't they discover the new number was higher right away? It's a thing that scientists are ashamed of - this history - because it's apparent that people did things like this: When they got a number that was too high above Millikan's, they thought something must be wrong - and they would look for and find a reason why something might be wrong. When they got a number close to Millikan's value they didn't look so hard. And so they eliminated the numbers that were too far off, and did other things like that...
This is why I'm an advocate of accepting/rejecting scientific papers based solely on methodology, with referees being given no information about the conclusions and with authors being forbidden from post-hoc tweaks. You do your experiment, and if you disagree with Millikan/conclude that ESP exists, so be it. Everyone is allowed to be wrong 5% of the time.
[1] I'm wearing my frequentist hat for the purposes of this post. Even if you are a Bayesian, you should still publish, however.