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:
[Event "EU-Cup 27th"] [Site "Rogaska Slatina"] [Date "2011.09.30"] [Round "6"] [White "Radjabov, Teimour"] [Black "Ponomariov, Ruslan"] [Result "1-0"] [ECO "C65"] [WhiteElo "2752"] [BlackElo "2758"] [PlyCount "145"] [EventDate "2011.09.25"] [EventType "team-swiss"] [EventRounds "7"] [EventCountry "SLO"] [Source "ChessBase"] [SourceDate "2011.11.11"] [WhiteTeam "SOCAR"] [BlackTeam "Tomsk-400"] [WhiteTeamCountry "AZE"] [BlackTeamCountry "RUS"] 1. e4 e5 2. Nf3 Nc6 3. Bb5 Nf6 4. d3 Bc5 5. O-O d6 6. c3 O-O 7. Nbd2 h6 8. h3 Bb6 9. Re1 Ne7 10. d4 c6 11. Bd3 Ng6 12. Nf1 Re8 13. Ng3 {"This position reminds me of typical set-ups in the Giuoco Piano. I have had a lot of games like this with the white pieces, so I usually prefer White here. If White manages to play d4 and keep the centre stable, then the long-term advantage should be on his side (a similar assessment would apply to Black if instead he had achieved ...d5 before White was ready with d4. According to this "rule" White’s position should be slightly better here." [Borki Predojevic]} Bd7 14. Be3 c5 15. dxe5 dxe5 16. Bc4 Be6 17. Qb3 Qc7 18. Nd2 Qc6 19. a4 Rad8 20. Qb5 Qxb5 21. Bxb5 Nd7 22. Nc4 Bc7 23. Red1 Ngf8 24. Nd6 Bxd6 25. Rxd6 Bb3 26. Nf1 a6 27. Be2 Re6 28. Bxc5 Bxa4 29. Rd2 Bc6 30. f3 Ree8 31. Bf2 Ne6 32. b4 Nf6 33. Bc4 Nf4 34. Ra5 Rc8 35. Ne3 Bd7 36. Rc2 Be6 37. Bf1 Red8 38. Rxe5 b5 39. Nf5 Bxf5 40. Rxf5 Nd3 41. Bb6 Rd6 42. Ba5 Nd7 43. Rd5 Rxd5 44. exd5 N3e5 45. d6 Rc6 46. f4 Ng6 47. c4 bxc4 48. Bc7 Nxf4 49. Rxc4 Rxc4 50. Bxc4 Kf8 51. Bxa6 Ke8 52. b5 Ne6 53. b6 Nxb6 54. Bxb6 Kd7 55. Bc4 Kxd6 56. Bxe6 Kxe6 57. Kf2 Kf5 58. Kf3 f6 59. Bd4 Kg5 60. Be3%2B Kf5 61. Bd2 h5 62. g3 Ke5 63. Bc3%2B Kf5 64. Bb2 Kg5 65. Ke4 Kg6 66. Kf4 Kf7 67. Kf5 g6%2B 68. Ke4 Ke6 69. Bc3 Kf7 70. Kd5 Ke7 71. Bd4 Kf7 72. Kd6 g5 73. g4 1-0" />
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