Why isn't my refrigerator water line delivering water?
Chances are you have a cheap saddle valve. Plumbers and appliance installers often use them because they're quick. They use a piercing needle built into them to punch a hole in copper pipes. The problem is that they're prone to corroding shut over time. Also, they have poor flow rates. 



You can simply close the old one and install a new one if you like, but evenutally it'll crud shut also, as the second one in my current home did. 

The right solution is to install a proper valve. This requires cutting pipe, but isn't terribly difficult. The benefits are longevity and flow, which is important when you have a drinking water dispenser and not just an ice maker.