I don't currently have a job at a "big game company" but I am looking for one and have interviewed to various extents with many of them: Bungie, Rockstar, Microsoft, Amazon, EA, Disney, etc. I have received one offer and am expecting to hear back from another of those companies tomorrow.
From the questions I've been asked, having book knowledge is, of course, helpful and will get you through a lot of the programming-specific interviews; to be a programmer, you're obviously going to need to know programmer things. Much of this can be learned by reading if that's how you learn. In every interview, though, I have pulled heavily from knowledge I picked up by doing various projects both professionally and as a hobby and being able to show things I've worked on—sometimes by pulling out my phone and showing a mobile game or by pointing them to my software development blog. My experiences have shown that interviewers are much more interested in what you have done than what you know. At a games programming panel at this year's GDC, the speakers all agreed that they want to see what you've done and don't care as much about schooling or book knowledge. Your mileage, of course, may vary.
As Superbest said, if you want to be an engine programmer, start programming engines. Similarly, if you want to be a gameplay programmer, write games and be able to show them off. They're very different disciplines that work together but are still quite different.
Additionally, there's more to being in a games programming job than knowing how to write code in a specific language. If your reading doesn't also include learning what makes games programming different from other kinds of programming, that may hurt you. If you can't demonstrate that you will mesh with the group, that may hurt you. In one interview, I spent over an hour doing linear algebra on a whiteboard, if you don't know the math that company is looking for, that may hurt you. In short, you're going to want to be well-rounded in ways that only reading programming books probably isn't going to enhance.