Most products depend on other products. e.g. Apple computers use components from other manufacturers.
Name one product that doesn't rely on another product.
Name one product that doesn't rely on another product.
That said, I think that: "don't make your service depend on other services without some sort of contract or agreement to provide" is a better rule of thumb than the one above.
Building your company around the core business of LinkedIn/Facebook/etc is dumb, because you are nothing more than a complement to their core mission. Best case, you become a commodity and achieve modest success, but the oxygen gets cut off at any time.. If you're very successful, you become a feature of the mother product.
When you're Apple buying memory from Samsung, that's different. Samsung's output is memory chips... that's the business. If nobody buys memory chips, they lose money.
Developing for the open web and trying to get people to websites in the current mobile/social era is increasingly brutal unless you are willing to be completely dependent on a handful of big companies.
Basing your product on top of LinkedIn? The core of their business isn't anywhere near providing that API.