In binary classification, this is basically a precision/recall trade-off. More precision (ie, fewer false positives) necessarily means less recall (ie, more false negatives). Classifier statistics like AUC basically tell you how good the classifier is in terms of how bad the trade-off is.
Adversarial situations - any time humans are trying to trick your classifier, in other words - will always make the trade-off worse.
There is a natural law in pattern recognition governing the split between sensitivity and specificity in scenarios with uncertainty, though I don't know if it has a special name. Catching more edge cases in a poorly-defined group requires allowing the inclusion of more things that look a lot like edge cases, but cases on the edge of a group are by definition similar to things outside of the group. The more you want to eliminate false negatives the more likely you are to produce false positives.