Real wage stagnation and union membership
My understanding is that theories and empirical methodologies demonstrating the relationship between declining union membership and wages risk overestimating the causality of the relationship.

Anyone who argues that businesses do NOT, in a relevant share of cases, endeavour to suppress union formation, activities or recruitment, whether individually or collectively, is either ignorant or a political shill. (Things are not nice 100% of the time in the other direction either.) This should be recognized, in order to proceed with rational discussion of the economics of the issue.

However, factors like increased international competition in lower-skilled jobs from a very large international labour pool (which is much more accessible to producers than in 1950 or 1980) can be one reason that the causality of declining union membership can be overestimated. 

Also, mechanisms of tacit collusion between businesses can act to hold down wages as well, for example with various agreements which prevent an employee from working for a competitor for some number of years after they leave (e.g., Google and Apple could both force employees to sign such an agreement as a condition of employment, thus holding down wages for both firms, which both account for a substantial share of employment in their specific areas of activity). This could be another explanation for some portion of wage stagnation that would be difficult to frame as resulting from declining union membership per se.

Between 1) international competition and 2) a proliferation of methods to suppress wages unrelated to union formation or membership per se, it should suffice to say that significant effort for robustness must be demonstrated before any empirical work can be taken seriously when discussing causality between declining union membership and 50 years of stagnant average wages.

The risk, then, is that an inadequate methodology which involves a) under-specification of a model, and/or which b) includes inadequate robustness checks, could indicate that declining union membership caused stagnant wages, when theoretically this could be a false positive when other factors explain both the declining union membership and also stagnant average wages.