The great tragedy of Science — the slaying of a beautiful hypothesis by an ugly fact. - Thomas Henry HuxleyAt times what science tells us is true (or more accurately what scientists put forth as theories that explain the data) conflicts with how nature appears to us. Even now there are still geocentrists because it "looks that way".
One striking example is dualism, the notion that the mind and body are separate. It's an intuitively powerful idea, I find at times I even catch myself as falling into the dualism trap. The reason I do this is because while intuitive, the success of physical models of the universe overrides the intuitive notion. In other words, the mind is a product of the brain and my experiences have a physical basis.
When an individual has an experience they say cannot be explained by evidence, does it mean that the laws of physics are ill-equipped to explain what happened? Or does it mean that the experience gives the impression of the impossibility of compatibility? This is an important distinction, it's the distinction between what is contradictory to our understanding of nature and what seems contradictory from the vantage point of the individual.
Take out-of-body experiences (OBEs) for example. This could be taken as evidence for dualism, because the experience of the brain leaving the body and giving a 3rd person vantage point shouldn't happen if the mind were physical. But such experiences can be seen as a product of brain activity, electromagnetically stimulate the right part of the brain and it can cause such effects. Perfectly explainable under a physicalist model of the universe, but to the individual profound evidence from experience.
If someone has a "spiritual" experience while taking LSD or peyote, surely the best explanation of the experience involves neurological reactions brought on by the use of chemical substances. That it's profound, that it is well beyond the usual experience, doesn't change that it's perfectly explainable in physicalist terms.
So many phenomena have at their heart a dualistic requirement. To name a few: astral projection, psychokinesis, ghosts, telepathy, communication with the dead, etc. Reports of experiences of such phenomena and others are common. Any of these if true would be powerful evidence against a physicalist notion of the brain.
This is where I feel the failure in taking experience over evidence. Something profound happens internally, yet the experience is biased by rationalisation in the brain. The one feels they were having an OBE doesn't mean they were literally having one, it's their internalised narrative to reconcile the experience. It's selling the interpretation with the experience!
Empirical inquiry is currently the best possible means to determine the validity of such claims. By taking the phenomena away from the internalised interpretation and testing to see how the claims stack up in a controlled environment, we are able to remove the subjective interpretation (from the subject at least) and make look to see if there is something really there. As yet, nothing.
People use this a reason to reject the scientific method, yet what it shows is a dissonance between comprehending the experience and comprehending the evidence. Denying the interpretation is not denying the experience, if it is profound then it doesn't need a profound explanation in order to be explained.
Qualia does seem in some philosophers' minds to pose a problem for physicialism if true. Whether or not qualia exist is of debate and one I'm nowhere near familiar enough with to comment on. I bring it up because at the core of qualia is subjective experience, and this is to contrast with the degree of experience which people use to question a pure physicalist reality.
If we can accept that emotional responses such as happiness and anger, or to feel pain or pleasure can be part of our reality, then it is talking about degrees of experience as opposed to just experience itself. If one can feel some sense of importance in a mundane situation (such as satisfaction at completing a task) then an overwhelming sense is but a stronger experience of the same phenomena. If one can feel a connection with another, then an overwhelming sense of love is but an extension of the same type of experience.
The point I'm trying to get across is the value of following evidence wherever it leads. From our vantage point it looks like the earth is at the centre of the universe. Everything rotates around us, and only through keen observation such anomalies such as the retrograde motion of the planets can be seen. Yet how else would it look to us? A heliocentric model of the solar system would have that same superficial appearance as a geocentric one, yet the heliocentric model can explain the anomalies which geocentrism can't. It just requires looking deeper.
That something has the appearance of being beyond the realm of the natural world, it doesn't mean it is. It's an argument from ignorance without so much as even trying to reconcile what is known. Until one takes it beyond the subjective, there's no way of differentiating between appearance and reality.