For what p-values can you say the data approached significance? I am in the field of clinical psychology, where statistical significance is commonly said compared to p = .05. For what values of p can I say data APPROACHED significance? 
 A: As already stated in comments: never. If you stick to Neyman-Pearson framework, then before doing your research you decide about some $p$-value threshold (like 0.05, or 0.001) and if get $p$-value that is smaller or equal to the threshold, then you decide to reject the null hypothesis. It doesn't make sense to say that the result is "almost" significant", the same as you don't say that you "almost passed an exam", or "almost survived a shot in a head". On another hand, if you stick to Fisherian framework, then you use $p$-values to measure the evidence against the null hypothesis, so you are not concerned with arbitrary thresholds. In either of the cases, the statement does not make sense.
Similar thing is said on the blog post recommended by conjugateprior in the comment:

You don’t need to play the significance testing game – there are
  better methods, like quoting the effect size with a confidence
  interval – but if you do, the rules are simple: the result is either
  significant or it isn’t.

(Check also the review of how people stretch the language to present insignificant $p$-values, as the results are quite entertaining.)
A: I believe that the convention is that p = .06 is considered "approaching" significance. Keep in mind though that it is not possible to say that it truly is approaching significance. Rather, this should give you reason to perhaps take a closer look at your data. Likely, the recommendation is to collect more data to see if an increase in power will give you a statistically significant result or not.
