I don't remember ever hearing the terms "Data Science" or "Big Data," but I do recall taking Department of Statistics courses with titles like "Data Mining," "Statistical & Machine Learning," and "Statistical Computing." We even sort of worked with what is now called "Big Data," by learning how to run large calculations in parallel using R's cluster computing packages.
As fancy and interesting as those courses were, I would only have a superficial understanding had I not also been exposed to more foundational/theoretical courses/topics like "Probability," "Measure Theory," "Mathematical Statistics," "Linear Regression," "Time Series Analysis," "Applied Stochastic Processes," etc.
When it comes to real-life practical implementation of all these ideas, it is necessary to have a pretty steep background in computer science. It's not enough to be able to do a couple runs in R or Hadoop. What is really called for here is at least an undergraduate level of knowledge in all the traditional areas of computer science, like programming languages, databases, computer systems, algorithms & data structures, etc.
Finally, the fourth ingredient is experience. The only way to really learn data analysis is through practice. It takes hundreds of hours of staring at data and code, struggling hard to find the relevant patterns in your data and improve the predictive performance of models. I guess this is the main advantage of pursuing a modern "Data Science" graduate degree, presumably you will spend a lot of time practicing data analysis on "real" data.
What I think irks people about the Data Science trend is that there seem to be a lot of people out there saying that you don't really need to be educated in mathematics and statistics. It's like saying that a mechanical engineer just has to know how to use 3D modeling computer software and doesn't need to know physics or mathematics ... that would lead to disastrous outcomes.