How should I interpret a journal rejection of "not of sufficient interest" or "does not meet journal standards"?
This is what happened to me particularly. Papers were rejected for those reasons and the journal never told me that they had found any error or that anything was wrong or inconclusive.
When a paper is rejected, do reviewers let you know if they found any error or they will never tell you even if they found one?
When a journal rejects a paper because it is not of sufficient interest for them or it does not meet the standards of the journal, it generally means exactly what they said. They thought the paper wasn't interesting or important enough.
Journals vary enormously in how selective or prestigious they are. Some journals will accept any paper that seems to be correct, original, and at least somewhat interesting. If a journal like that rejects a paper for this reason, then either it's not a very good paper, or they are being unfair (perhaps accidentally - peer review isn't perfect, so occasionally you just have to try again), or you submitted it to the wrong sort of journal (the line between different subfields can be blurry, but if you choose the wrong side it decreases your chances of acceptance).
Other journals impose extremely high standards and only want to publish papers on exciting breakthroughs. In that case, there might be nothing wrong with the paper at all, and the only issue is that they have received other submissions they like even better.
And, of course, there's a whole range of journals in between these extremes. Depending on where the journal is in this range, it will shift from a statement about your paper to a statement about the journal's high standards, and there's no way to be more precise without knowing more about the situation.
The best source of advice is a trusted mentor in your field, but I wouldn't get too worried on the basis of one rejection. Look over the paper again with fresh eyes, make sure the introduction and conclusions are compelling, choose another journal, and resubmit. If you run into this problem repeatedly, then something's wrong (either the paper needs work or you need to choose more appropriate journals) and that would be a good time to seek more detailed advice.
The short answer is that if a reviewer found an error, the reviewer will generally tell you. If you get a rejection without any further comments, the likely reason is that the reviewer read the outline and main results, and concluded that it wasn't necessary to go through the paper carefully to decide to reject it, probably because the results weren't significant enough for that journal. (Some journals specifically request that reviewers do a quick read of the article within a couple weeks of receipt, to see if it has any hope of being published; it often takes reviewers months to do a full read through, and if it has no chance, it's kinder to the author to give a quick rejection so the author can promptly resubmit to a journal which might publish it.)
However sometimes the reviewer has carefully read through the paper before recommending rejecting it, and in that case the reviewer usually (at least in my experience) includes a list of suggestions or comments (including pointing out any errors the reviewer found).
Often, when a paper is rejected by the editor (without being sent out for review), the standard response is along the lines of "not of sufficient interest".
How should you interpret this? One interpretation is to be humble and select another journal that will be interest. Another interpretation is that good and interesting science has been poorly presented.
One possibility that has not yet been covered is that the paper did not present interesting science in an interesting way. I had a paper that was rejected from a number of journals although I felt strongly that these judgements were inconsistent with the work that I had done. I also recognized that such quick judgments can be based on the title and abstract, and on re-reading these, I realized that the main, important points of my paper were not given enough emphasis up front. After minor revisions to focus my readers, my next submission was very warmly received.
