I suspect this is a characterized event but I don't know what it is called so I cant look it up. Thinking about it, it seems to confound the False Discovery Rate's (FDR) ability to estimate the background uniform distribution causing no values to survive the multiple test correction though there clearly appears to be a significant signal.
1) Is there a name for this phenomenon? What causes it? Is it enough to say that there are several inappropriate tests and can for that reason be ignored? Am I justified in trimming this long insignificant tail since these tests were clearly inappropriate?
2) Is there a multiple test correction robust to this phenomenon?
Is this a different phenomenon? Does it require a different response and/or multiple test correction in order to salvage the signal?