How can I do better research when the experimental design is lacking? I would like to do first-class research. However, how do I actually accomplish that when my supervisors design not-so-great experiments and I don't have the resources to do more independent work?

In many fields (e.g. psychology, neuroscience), graduate students work in a group (i.e. lab) studying topics that their advisor finds interesting. Each student might have their own thesis topic, but all of the work is focused on the same big question or general research topic.

Typically the data collection is mostly dictated by the senior researchers, as it is very expensive. Once it is collected, the lab might spend several years analyzing it, especially in bigger projects. This means that the potential of a project is mostly determined by the experimental design. If there is something wrong with it, not much can be done afterwards and the results must still be published.

So what can a young researcher do in these fields if the experimental designs (decided on by the advisor or other senior researchers) are lacking? Especially if I have done most of what I think is possible post-hoc (e.g. with the statistics)?
Though I am not familiar with the specific field you are referring to (neuroscience &amp;c.) one big question presents itself:

Did you actually talk to your adviser/other students about the "problems" you perceive in the experimental setup? Are these actually "problems" or maybe (just maybe) an insufficient understanding on your part?

If it is indeed a problem of experimental design I would propose to contact the ones responsible for the design and try to work something out - i.e., additional experiments or (if possible) looking for external data.