I think that, for the majority of students, your advisor's advice is correct. You need to focus on a particular problem, otherwise you won't solve it, and you can't expect to learn everything from text-books in advance, since trying to do so will lead you to being bogged down in books forever.
I think that Paul Siegel's suggestion is sensible. If you enjoy reading about different parts of math, then build in some time to your schedule for doing this. Especially if you feel that your work on your thesis problem is going nowhere, it can be good to take a break, and putting your problem aside to do some general reading is one way of doing that.
But one thing to bear in mind is that (despite the way it may appear) most problems are not solved by having mastery of a big machine that is then applied to the problem at hand. Rather, they typically reduce to concrete questions in linear algebra, calculus, or combinatorics. One part of the difficulty in solving a problem is finding this kind of reduction (this is where machines can sometimes be useful), so that the problem turns into something you can really solve. This usually takes time, not time reading texts, but time bashing your head against the question. One reason I mention this is that you probably
have more knowledge of the math you will need to solve your question than you think; the difficulty is figuring out how to apply that knowledge, which is something that comes with practice. (Ben Webster's advice along these lines is very good.)
One other thing: reading papers in the same field as your problem, as a clue to techniques for solving your problem, is often a good thing to do, and may be a compromise between working solely on your problem and reading for general knowledge.
I can't answer this question, but perhaps can offer some info:
Edit 1: I realized that what appears below would perhaps be more suitable as an answer to a question like: "How do you cope with the reality of being at an undergraduate-focused school when you had your heart set on a research career?" This is markedly different from the OP's question, but evidently my answer seems useful for this one. Perhaps I've implicitly answered the question...It would be nice if this question received an actual explicit answer! (e.g. I think Joseph O'Rourke's deleted answer was closer to the mark than mine is!)
1') Important Edit 2 Recent economic pressures on undergraduate colleges connected with projected enrollment decline, health insurance laws, public perception of higher education and resulting changes in "product delivery", has led me to revisit the advice I gave in (1) below. It is very unlikely that any but perhaps a few elite undergraduate liberal arts colleges will be able to afford to support research substantially in a sustainable way. Small class sizes and reduced teaching loads are regarded by most colleges as mutually exclusive and unaffordable to maintain simultaneously. Since the former is a big selling point of the liberal arts college, the latter probably will not easily be found at a liberal arts college without financial problems at the institution following closely behind, if the institution is one of the many that are primarily enrollment-driven. If you happen to enjoy research and don't want to be forced out of it by economic reasons like teaching load or salary that needs supplement, do not look for a job at a liberal arts college with less than a billion dollar endowment. (I believe this edit is important, since the old answer might mislead future job candidates for such institutions…which I really do not want to do.)
1) Not all liberal arts schools are the same. Not considering the elite schools, some departments have 3 hour loads distributed over two courses. I work at such a place. Office hours and service are still more of a commitment than at a research school, but there is still time left over to work. My college is clearly trying to raise its research profile a little. My feeling is that due to the sheer number of qualified PhDs looking for jobs this is an emerging market, and so colleges that want to move up in the rankings will consider reducing teaching loads in order to take advantage of the oversupply of strong research candidates. You won't be able to write two good papers per year...perhaps not even one good paper per year...but you will be able to do research at such a place. Chances are, if you have tenure at a small school then you have a knack for engaging undergraduates (whether you like it or not). I'll bet you will be an attractive candidate for a move to such a department.
2) This is presumptuous to write, because it's pretty obvious, but getting a good research job usually requires making very serious sacrifices. Many of my research friends still are working hard on work/life balance issues. Some have had to be separated from a spouse (often on a different continent) in order to try to get a decent research job. I can't help but think that people who land these jobs often quite deserve them. I can say that my own unwillingness to expose myself to such risk is the main reason why I originally accepted a job at a liberal arts college instead of trying to go to a research university.
3) Having just chaired a search committee, I've recently reviewed more than 750 applications for a tenure-track liberal arts job. The gut-wrenching truth is that there were more excellent research candidates than the market can support. If we restrict our attention only to those with at least one postdoc, it is probably possible to fill all openings on MathJobs with candidates having more than 5 publications (and this is conservative). Budgets allow for bringing a few candidates to campus for an interview, and almost nothing is worse for a potential employer than blowing a campus interview on someone who is not serious about a position. To some extent, your vulnerability as an applicant is attractive to the interviewer. If you have tenure already, it is possible that you will be harder to secure...or worse, will be seen as collecting offers to bargain for a raise at your current job. Unfortunately, I have strong evidence that this happens. In one case, this evidence stopped us from talking to an untenured candidate. I think that this is the main reason why it is nearly impossible to move after tenure. (Sorry to kill hope...I hope I am wrong about this!)
4) One final important point I forgot to include before. As you mourn the fact that you never will teach graduate courses or supervise PhD students, keep in mind that there is an important positive side to this fact: You will not have to help such PhD students find an academic job in the current market. I imagine that having taken a job at an undergraduate institution, if you were to relocate to a more research-based position it probably will not be Princeton. Instead, you will probably find yourself at a lower-tier research school. I speculate that PhD students graduating from such schools are at a disadvantage for landing excellent postdocs (If I'm wrong about this, someone please contradict me) and subsequently finding permanent academic employment. I imagine watching students go through this would be very stressful for a conscientious thesis advisor.
Best Answer
I may have spent enough time feeling frustrated that I can't find anything that resembles this quote in Google Books. :-) The closest that I came was a quote from Halmos in the book "Paul Halmos celebrating 50 years of mathematics". Halmos explained that he started walking a lot on his doctor's advice, as an alternative to hypochondria. This wasn't about walking in relation to research.
Are you sure that it wasn't e-mail?