General Case:

In an otherwise well-constructed discussion on a subject, the author says something that reveals a significant misunderstanding of the subject, casting doubt on the entire piece, and the ability of the author to think about it sensibly.

Motivating Example:

A few years ago, a lot of public libraries in the UK were closed under austerity measures. Author Philip Pullman (a highly-educated, eloquent and thoughtful man) gave a speech on the subject, which was transcribed and widely circulated online. It was about the non-pecuniary value of libraries, and their value as educational and community resources. It was a very strong speech, but at one point it put forward the proposition that the value of libraries are completely incalculable and beyond measure. This took the wind out of the speech's sails for me, and my takeaway was "you write and speak very well, but you clearly can't be trusted to think about this subject in any useful way".

I experience this quite a lot. I'll be reading something online, mentally nodding along, thinking "yeah, this makes sense", and then the author will undermine all their credibility, not by saying something radical or obnoxious or unworkable or ignorant, but by saying something that demonstrates they don't know how to think properly about the issue.