The key is confirmed experimental results that are other than predicted by established theory. When theory is very well established, there is a tendency to out-of-hand dismiss contradictory results as probable errors. Sometimes that "theory of error" is accepted without the errors ever being identified. This especially can happen if there is mixed success in confirmation, which can happen when a phenomenon is not understood and is difficult to set up.
Nuclear physics is such a field, where quantum mechanics is incredibly successful at making accurate predictions when the environment is simple, i.e., in a plasma.
However, in the solid state, to apply quantum mechanics, to predict fusion probabilities, notably, requires simplifying assumptions.
Seeking to test the accuracy of these assumptions, Pons and Fleischmann, starting in about 1984, found a heat anomaly. The effect was difficult to set up, it required loading of deuterium into palladium at a ratio higher than was normally considered possible, and most palladium samples didn't work.
They were not ready to announce the work, but the University of Utah forced them, for intellectual property reasons, to hold a press conference. All hell broke loose, it is said that for a few months the bulk of the U.S. discretionary research budget was spent trying to reproduce their results.
Most of these efforts were based on inadequate information about the original research, most failed (for reasons that are now understood), and a cascade developed that there was nothing but incompetence behind the finding.
However, some researchers persisted, and eventually there were many independent confirmations, and the heat effect was found, by a dozen research groups, to be correlated with the production of helium, at the ratio expected for deuterium fusion to helium, within experimental error. Helium was not expected to be a normal product of deuterium fusion (it's a rare branch), and when normal (hot) fusion does result in helium, there is always a gamma ray, required by conservation of momentum. No gamma rays.
The mechanism is not known. What I've written here is what you will find if you look for recent reviews of the field in mainstream journals. (See especially Storms, "Status of cold fusion (2010)," Naturwissenschaften.)
But the opinion is still extremely common that the whole thing is "pathological science," or worse.
Until the mechanism is known, this might be a laboratory curiosity, or it could open up a whole new territory, with vast implications. More research is needed.
Some time ago I learned of the metaphor of 'digging the bull's horn'. This might sound a little strange, since horns are mostly hollow, but imagine a bull's horn used to store black powder. In the beginning the work is easy and you can scoop out a lot powder with very little effort. As you dig down, though, each scoop yields less powder as you dig into the narrow part of the horn until the only way you can get out more powder is to turn the horn over a dump it out.
It's often the same way with learning. When you start out in a subject there is a lot to be learned (both in quantity of material you have not yet seen and in quantity of benefits you have to gain from the information), but as you dig deeper into a subject the useful insights come less often or are more limited in scope. Eventually you dig down so far that the only way to learn more is to discover new things that no one has yet learned (to stretch the metaphor, you have to add your own powder back to dig out).
It's useful to know that you're digging the bull's horn when learning because, unless you really enjoy a subject or have some reason to believe that contributing to it is worthwhile, you can know in advance that most of the really valuable insights you'll gain will come early on. If you want to benefit from knowing about as much stuff as possible, you'll often want to stop actively pursuing a subject unless you want to make a career out of it.
But, for a few subjects, this isn't true. Sometimes, as you continue to learn the last few hard things that don't seem to provide big, broadly-useful insights, you manage to accumulate a critical level of knowledge about the subject that opens up a whole new world of insights to you that were previously hidden. To push the metaphor, you eventually dig so deep that you come out the other side to find a huge pile of powder.
The Way seems to be one of those subjects you can dig past the end of: there are some people who have mastered The Way to such an extent that they have access to a huge range of benefits not available to those still digging the horn. But when it comes to other subjects, how do you know? Great insights could be hiding beyond currently obscure fields of study because no one has bothered to dig deep enough. Aside from having clear examples of people who came out the other side to give us reason to believe it's worth while to deep really deep on some subjects, is there any way we can make a good prediction about what subjects may be worth digging to the end of the bull's horn?