'Black Swans' are extraordinary, highly unexpected events. Inspired by Nicholas Taleb's 2007 book of the same title, my contention in this paper is that exceptional events carry a great - and as yet untapped - explanatory potential in the field of archaeology. Archaeological exceptions, i.e. instances or occurrences deviating from the archaeological norms and patterns as constructed in our discipline, are usually dismissed, ignored or deemed irrelevant to any account of social dynamics and human behaviour in the past. Upon closer examination, however, archaeological 'Black Swans' may provide hitherto unsuspected explanations for certain phenomena of past societies. The above claim will be illustrated here on the basis of two examples from the mortuary record of south-eastern Iberia's Argaric society (c. 2250-1450 cal. bc). In the first case - studied by Colomer in her PhD dissertation - we will see how substandard funerary urns that try to replicate typical funerary containers may give us the clue to understand pottery manufacture in the Argaric world. In the second instance, we shall explore, through the study of a singular silver awl found in an Argaric tomb, the symbolic value that some objects related to daily maintenance chores may have attained as grave goods.