For example, I used to be too narrow in my opening study methods and, like many players when they are first exposed to opening theory, focused on choosing and memorizing "book" variations. Study of complete, annotated games featuring chosen variations was a big step up from this constrained and ultimately counterproductive method, and I believe this should be the cornerstone of most improving players' practice. Another step forward in knowledge and sophistication, however, is looking at a broader range of games in order to take away valuable lessons at a conceptual level.
Naturally you need to have a balanced approach to selecting games to study, since with a limited time budget you can't just take in everything indiscriminately. However, while looking at games from contemporary tournaments of interest, or when working through collections of annotated games of world-class players, similar concepts across different openings should pop out at you during the process.
Sometimes I find that identifying an analagous concept in a completely different opening has an even greater impact on my understanding of it, perhaps due to its unexpected nature. These types of common concepts are, by definition, worthy of further study and examination due to their appearance in multiple types of games. Striving to understand small differences in how concepts are applied across different games, or in different variations, I believe is also one of the keys to mastering positional understanding. (See Training quote of the day #2)
Examples of openings cross-training are legion; here are some that I have run across at various points in my studies. As can be seen below, cross-training opening ideas can range from direct transpositions between openings, which are more obvious, to individual maneuvers that can be applied in similar circumstances.
Caro-Kann / Slav combination: I've played both openings for a long time and I don't believe they have a large number of identical ideas, despite the duplication of Black's first two moves (1...c6 followed by 2...d5) and the common initial idea of supporting the d5 pawn in the center. Their main commonality is that they both normally lead to semi-open type games, rather than open or closed positions. Nevertheless, there are some benefits to being able to play both, given some early transposition possibilities (as occurred in Annotated Game #67). Switching to the Caro-Kann is also an easy way to meet the Blackmar-Diemer Gambit (1. d4 d5 2. e4 c6), if you have it in your repertoire.
Caro-Kann / Queen's Gambit Declined: further along in the opening, the Caro-Kann can in some cases also merge with queen's pawn openings, as occurs in a popular line of the Panov-Botvinnink Attack after Black plays 5...e6; see Annotated Game #38 and Annotated Game #123 for personal examples.
Slav / Stonewall Dutch: the Stonewall can be reached from a Slav move-order and can be used to good effect that way, as Anna Zatonskih did in the 2013 U.S. Championships.
Bishop retreat to h2/h7: an example of the value of this maneuver for White can be found in this 2014 U.S. Championship game by Gata Kamsky featuring the London System, while its analog for Black can be found in the analysis to Annotated Game #124. In both cases the point is to preserve the light-square bishop rather than allow the opponent to trade it off.
And finally, here is a more sophisticated example, with the comment excerpted from the November 2011 Chess Evolution analysis of a top-level game in the Berlin Defense to the Ruy Lopez:
As a postscript, a couple of other observations on opening cross-training from the chess blogosphere:
GM Nigel Davies - The Benefits of Cross Training
GM Vinay Bhat - Mind = Blown
Sputnick - Responding to 1.d4 with the Nimzo-Indian and Ragozin