There is no single measure for "robustness" and a lot of it boils down to the field. Usually physics has a very high degree of precision: p value close to 0. For instance, the Higgs boson was a confidence level of 5 sigma, while biology or psychology is going to usually have more leeway: maybe 2 sigma (Sigma).
However, there should be no (statistical significance) criterion to publish. The idea that only positive studies should be published has led to biases in available data in a number of fields (Fighting publication bias: introducing the Negative Results section). Negative results are very useful to us as robustness of a theory is often evaluated using meta-analyses. If a person fails to publish their results because it does not robustly justify their theory, then the data available to perform a meta-analysis is diminished.
Clearly, you want the study to be properly run and you want the method to be reasonable. A garbage study, regardless of the statistical significance measured, is going to be looked upon negatively.
That being said, if your topic is something that has been researched a number of times, a small study is not going to be looked at very well, as there is already "better" information available. The study should provide some kind of new insight into the topic.