Google has over 200 signals it uses to rank results. Given Google's legendary PageRank algorithm, based on links, it has led to a lot of people worrying about links way too much. That's not to say quality links aren't still important, but just because you have a whole bunch of links, it doesn't mean your site is going to rank well.
Google's Matt Cutts posted an interesting webmaster help video under the title: "Will Google Provide More Link Data For All Sites?" It's Cutts' response to the user-submitted question:
In the wake of the demise of Yahoo Site Explorer, does Google Webmaster Tools plan to take up the reigns this product once provided to SEO's everywhere?
Cutts responds, "What I think you're asking is actually code for 'will you give me a lot of links?' and let me give you some context about Google's policies on that. I know that Yahoo Site Explorer gave a lot of links, but Yahoo Site Explorer is going away. Microsoft used to give a lot of links. And they saw so much abuse and so many people hitting it really, really hard that I think they turn that off so that people wouldn't be tempted to just keep pounding them and pounding their servers."
"So our policy has been to give a subsample of links to anybody for any given page or any given site-- and you can do that with a link colon command--and to give a much more exhaustive, much more full list of links to the actual site owner," says Cutts. "And let me tell you why I think that's a little bit more of a balanced plan. Yahoo Site Explorer, they were giving a lot of links, but they weren't giving links that Google knew about. And certainly, they don't know which links Google really trusts. And so I think a lot of people sometimes focus on the low-quality links that a competitor has, and they don't realize that the vast majority of times, those links aren't counting."
"So, for example, the New York Times sent us a sample of literally thousands of links that they were wondering how many of these count because they'd gotten it from some third party or other source of links," he adds. "And the answer was that basically none of those links had counted. And so it's a little easy for people to get obsessed by looking at the backlinks of their competitors and saying, 'oh, they're doing this bad thing or that bad thing.' And they might not know the good links. And they might not know that a lot of those links aren't counted at all."
"So I also think that it's a relatively good policy because you deserve to know your own links," he continues. "I think that's perfectly defensible. But it doesn't provide that much help to give all the links to a competitor site unless you're maybe an SEO, or your a competitor, or something along those lines. So for somebody like a librarian or a power searcher or something like that, using link colon and getting a nice sample, a fair fraction of links to a particular page or to a particular website, is a very good policy."
"I think that's defensible, but I don't expect us to show all the links that we know of for all the different sites that we know of, just because people tend to focus on the wrong thing," he concludes. "They don't know which links really count. So they tend to obsess about all the bad links their competitors have and only look at the good links that they have. And it's probably the case that surfacing this data makes it so that you're helping the people who really, really, really want to try to get all their competitors backlinks or whatever. And I just think it's a little bit more equitable to say, OK, you're allowed to see as many of the backlinks as we can give you for your own site, but maybe not for every other site. You can get a sampling, so you can get an idea of what they're like, but I wouldn't expect us to try to provide a full snapshot for every single site."
Links obviously aren't everything, and if you follow Google's changes, it's easy to see that other signals have been given a lot more significance in recent memory. This includes things like content quality, social signals and freshness. If you're that worried about the number of links you have, you're living in the wrong era of search.
Granted, links have value beyond search ranking. They still provide more potential referrals to your site, but in terms of Google, the search engine is moving more and more away from the traditional 10 organic links anyway, with more personalized results, fresher results, blended (universal search) results, and more direct answers.