How long will it take to completely run out of power a Li-ion battery of 3.7V and 1000mAh, if the load consumes 16mA?
You're not doing anything wrong. In fact, you're doing exactly right by not stressing your battery. You've run across a couple of things. First, Ah capacity is always sized somewhat conservatively, so that people don't get disappointed and try a competitor's battery. This is counterbalanced by the desire to claim as much capacity as possible so people don't go buy a competitor's battery. The other factor is that, as current decreases the efficiency of the battery increases. With low current demands the internal chemistry can produce more power with, among other things, less wasted heat. 

I'm a bit surprised that your efficiency gain is this large, though.