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. NSum stops when the error estimates it gets imply that either the accuracy or precision sought has been reached.
You should realize that with sufficiently pathological summands, the algorithms used by NSum 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 NSum.
VerifyConvergence is only used for sums with infinite limits.