So over the course of the last years, I've developed a certain knee-jerk reaction where I'm super cautious about p-values between (~).01 and .05.>
I've been working with a few large-scale data sets and one thing I've experienced is that effects that are robust and are likely to pop up in other data sets as well come with much lower p-values.>
Example from our specification curve paper (home.uni-leipzig.de/diffdiag/pppd/…): I get super tiny p-values, almost regardless of how I slice and dice my data.> 
Now in contrast, if there's nothing going on, there is nothing going on. But notice that the "just significant" values just under .05 occur just as often as they do when there is strong evidence for an effect.> 
So if I run some analyses and end up with p = .03, I'll initially assume that I've encountered the 2nd situation (nothing going on) and double check whether the "finding" holds up or replicates in another sample.>
That's obviously just an intuition, there are formal arguments to be made here as well, and things look quite different if the analytic plan has been exactly pre-specified! But I still consider it one of the more helpful gut feelings.>
And needless to say, smart people such as @siminevazire have already blogged about this: sometimesimwrong.typepad.com/wrong/2015/06/…
