What are the long term effects of Organic Surface Protection (OSP) on PCBs?
I guess your doubts arise from the look of the PCBs, where the copper may look like it hasn't been treated at all.

This document Precoplat refers to on its website says  


  The varnish forms an airtight cover to the copper surface and “allows” the processed boards a storage life of 6 months maximum. Increasingly, after this time, the encapsulated surface protection characteristic is gradually lost.


So your boards should be safe for half a year, which sounds reasonable. (If your PCBs remain in storage longer than that between production and assembly it looks like you're having a logistics problem.)  

Areas which don't receive solder (paste) before soldering are no longer protected after the soldering, since the OSP layer breaks off at temperatures above 150°C (same document).