As is the case with many truisms, this statement has a kernel of truth but can also be quite misleading. Assume, for example, that you are applying for a job as a brain surgeon or an engineer or a car mechanic. While having an extended network of introductions may help get you a first interview, in order to actually get a job and keep it, you need to have the appropriate knowledge and technical skills. You won't get very far in most jobs without the requisite skill set. Of course, keeping a job also requires that you can work well with colleagues and customers, and having a good network will be an advantage in many aspects of your career.
Even in jobs such as sales, which rely heavily on people skills, increasingly one needs technical skills such as knowing how to create blogs in WordPress, using spreadsheets, and knowing something about how to use big data and social media. While having a wide network of acquaintances can be invaluable, introducing you to people who can help you find information as well as jobs, it is also important to have a base of technical skills.