Headphones

Fancy materials and weave patterns can definitely have measurable effects. Everyone who's done some physics or electrical theory training knows that. It's just that the differences aren't significant in the context of a headphone listening situation. But many manufacturers like to blur the lines on that.

As other commenters have mentioned, all cables aren't perfect just because there is an upper limit of useful performance. You can definitely still find cables that will perform more or less sub-optimally depending on the application. Back in the olden days, rubbish cables may have been more proliferated.

Placebo is an important factor when dealing with subjective or intangible things. Add on some cultural prestige (being in the right in-crowd), and it gets messy. The fact of the matter is that if we listen to two cables that are identical, or create a tiny difference through impedance or wherever, then we will hear the difference that we expect or want to hear.

But there is also the feel-good factor. If getting a cable that is made of exotic materials and is designed and made well makes you enjoy listening more, then it has arguably done its job. Just like a comfy chair makes the music sound better, without actually changing the sound. As long as you don't tip over into delusion and total financial irresponsibility, then I don't think there's anything inherently wrong with it.

In summary, it's not as simple as many people would like to believe, and there are cores of reason to all delusions. You mentioned burn-in and it might very well have mattered on older transducer designs. It's certainly something you might need to deal with on vacuum tubes.