[dwaltrip]

Are negative results more likely to have methodological or experimental error than positive results? I've never heard that claim.

I would actually expect the opposite, as finding evidence to support a new claim should be more difficult than not finding evidence.

My understanding is that a negative result has the following format: "we did such and such experiment and did not find a significant statistical relationship between thing 1 and thing 2, after controlling for a bunch of other things".

It's worth noting that this by no means proves that there isn't a relationship, it just means that study wasn't able to find evidence of one. It could be a piece of the puzzle for a potential strong case against such a relationship, or that further research is needed to untangle any confounding factors. Which is why I think all methodologically sounds results should be published, no matter how unflashy or boring.