You run the risk of sacrificing depth for breadth. With finite educational resources (time and money in real dollars that are likely borrowed), most people are attempting to maximize their earning potential post-graduation.
Part of the problem is that profit seeking companies don't favor the renaissance man / woman because they need people down in the trenches, implementing the mundane details of the grand ideas dreamt up by innovators. The reality is that most people are not a da Vinci or Jobs (both would have made significant contributions to society without diverse educations--there's likely no shortage of examples of hyper-focused authors, scientists, artists, etc. who were equally successful).
Although I'm not a huge fan of argument by personal anecdote I think it applies here. As an undergraduate engineering student, I spent a large portion of my educational resources, 20-50% of my time / energy, on policy debate [1]. Each year, I intensely researched, organized, presented, and actively debated international politics, economics, philosophy, sociology, and literary criticism in the context of a specific topic. At the conclusion of each year, I could have written a thesis on a relevant subset of each topic--I say this as someone who has written and defended an engineering thesis. Post-graduation, however, I find myself working for an engineering firm, doing engineering work. During the interview and continuing to this day (4+ years later), no one cares how much I know about West Africa agricultural problems, biopolitics, feminism, Ex parte Quirin, or the AUMF. It's simply not relevant to work I do. I would love a job that uses all of my interests and talents, but, especially in this case, that seems unlikely. "Looking for candidate with strong background in Structural Dynamics, Aeroelasticity, Feminism, and Middle Eastern politics!" Maybe I'm just not creative enough.
Am I well read? Sure. Do I sound intelligent over a beer or two? Absolutely. Am I a contributing member of a well-informed electorate? You betcha! But would that time / effort been better spent on engineering-specific extracurricular activities? I know what the hiring manager at my company and my boss would probably tell you.
Sure I retained the research, organization, presentation, and public speaking skills, but they're really only marginally relevant to my current work. I also would have probably developed those skills if I was involved in DBF [2].