With the Euler-Maclaurin method, the options AccuracyGoal and PrecisionGoal can be used to specify the accuracy and precision to try and get in the final answer. NProduct stops when the error estimates it gets imply that either the accuracy or precision sought has been reached.
You should realize that in sufficiently pathological cases, the algorithms used by NProduct can give wrong answers. In most cases, you can test the answer by looking at its sensitivity to changes in the setting of options for NProduct.
VerifyConvergence is only used for products with infinite limits.