Published studies using functional and structural MRI include many errors in the way data are analyzed and conclusions reported. This was observed when working on a comprehensive review of the neural bases of synesthesia, but these errors are probably endemic to neuroimaging studies. All studies reviewed had based their conclusions using Null Hypothesis Significance Tests (NHST). NHST have yet been criticized since their inception because they are more appropriate for taking decisions related to a Null hypothesis (like in manufacturing) than for making inferences about behavioral and neuronal processes. Here I focus on a few key problems of NHST related to brain imaging techniques, and explain why or when we should not rely on significance tests. I also observed that, often, the ill-posed logic of NHST was even not correctly applied, and describe what I identified as common mistakes or at least problematic practices in published papers, in light of what could be considered as the very basics of statistical inference. MRI statistics also involve much more complex issues than standard statistical inference. Analysis pipelines vary a lot between studies, even for those using the same software, and there is no consensus which pipeline is the best. I propose a synthetic view of the logic behind the possible methodological choices, and warn against the usage and interpretation of two statistical methods popular in brain imaging studies, the false discovery rate (FDR) procedure and permutation tests. I suggest that current models for the analysis of brain imaging data suffer from serious limitations and call for a revision taking into account the new statistics (confidence intervals) logic.