From what you describe, it sounds like flux residues that are being left on the pcb. This can occur from either; too high a reflow temperature causing the flux to burn and contaminate the pcb or too low a reflow temperature causing the flux to wet to the board but not allow the solder balls to melt.

You first order of finding a solution would be to use an optical inspection system like the VPI-1000 from Easy Braid Corp., It will allow you to look under mounted BGAs and determine what the material is.

Contaminants on the BGA balls (left behind on the balls due to due to improper cleaning during the BGA mfg process) may be interfering with the solder paste during the SMT process; and causing the opens.

There have been instances where plain tap water has been used to clean the BGA instead of DI water - during SMT reflow, these contaminants reacted with solder paste to form solid "gunk" and caused opens.

BGA ball cleanliness needs to be verified.

Also a BGA mfg process that uses a no-clean process could result in a wider process window because no contaminants are introduced due to a cleaning step.

SEM-EDS & Raman Spectroscopy can be used to analyze the composition.

The answer to the question is the development of an analytical test protocol.

1. The substance needs to be identified – if it is elemental then a SEM / EDX needs to be performed to determine the elemental composition.
2. The substance needs to be tested for organic composition as well – This is done with an FTIR to determine the organic spectrum. This will then be put against other known spectrums to give a percent confidence of what the organic substance is or atleast what family it is in – ie. Flux residue, uncured epoxy resin, etc. This gives one a good idea of where to look for root cause.
3. Then once identified the root cause or source of the contaminate has to be determined and why it is there and effecting the solderability of the bga balls.
4. Then one needs to replicate the problem to insure all variables are understood and then a corrective action will be developed to insure the problem is eliminated or atleast controlled depending on the real problem.

The issue could be excessive flux residue or uncured epoxy resin. A good test protocol will identify the substance and once identified then root cause can be determined.

STI Electronics performs this type of failure analysis work daily and has the tools to find root cause.

I hope this helps the company in need. Good luck.