How do we test reliability and quality so as to minimize the risk of board failure in the field?
There are several different ways to approach this problem.  Typically one does testing where  the device is operated under stressful conditions to reduce its lifetime.  This can include elevated temperature, temperature cycling, vibration, humidity, etc.  Sometimes the test protocol runs to failure.  The failure may be repaired and the test resumed until the next failure, etc.  Often many samples are run at the same time.

For more information see:
http://en.wikipedia.org/wiki/Highly_accelerated_life_test

There are many companies which specialize in this type of testing service.  I suggest that you contact one of them.