I was clicking around Simply Statistics today when I came across a quote from a post by Jeff Leeks that is a great analogy for how I think about statistical analysis. (And my apologies to Jeff Leeks, because I'm probably taking my interpretation further than he intended.)
"But it also means we need to adapt our thinking about what it means to teach and perform statistics. We need to focus increasingly on interpretation and critique and away from formulas and memorization (think English composition versus grammar)."
The emphasis is mine. I think that most would agree that knowing lots about grammar, conjugation, and how to string words together into sentences are necessary but insufficient to being a good writer. And while there are specific skills to make something you write more readable, great writing involves creativity, thinking outside the box, or at least thinking like someone sitting in a different box.
I think statistics can be similar, at least in the social sciences (where most of my experience lies). Time and again, it's a little creative stroke that moves the analysis from "technically executed" to actually speaking towards the phenomena I'm studying. Often times, that creative stroke comes from having seen a problem or an approach from another discipline and/or context.
One may argue that, "we can teach you how to write, but we can't teach you to be a great writer." And so it goes with statistics. But I believe the key is breadth as well as depth. So, throw some history, political science, sociology, economics, etc. alongside your pile of math and comp. sci. books. At the least, the writing's probably better.