Richard,
 
First, I assume these are component holes?
 
If so, can you perform hand rework until the cause is identified and resolved?
 
If you can't then I'd agree it is a huge problem.
 
4 - 5 holes per board, okay are these in the same general area? Widely scattered?
 
Is the entire hole non-plated? Has the copper wetted at all?
 
Do you have extra boards? What does the copper look like on those? Is it discolored or uneven in appearance?
 
Can't say with certainty that it's one problem, not with the info provided. Also, can't tell you how to solve it for the same reason.
I'll try to assist but I need more info, see if you can provide answers to the questions above, I'll do what I can.
 
Regards,
 
Franklin D Asbell
Network Circuits, Inc.
Irving, Texas