Quarters of nobility is an expression used in the bestowal of hereditary titles and refers to the number of generations in which noble status has been held by a family regardless of whether a title was actually in use by each person in the ancestral line in question.
For example, a person having sixteen quarterings (formally in heraldry Seize Quartiers) might have exclusively noble ancestry for the four previous generations (i.e., to the great-great-grandparent level): Given two parents per generation, four generations of uninterrupted nobility = 24 = 16. Alternatively, such a person might have exclusively noble ancestry for the five previous generations on one side but have a commoner for his or her other parent, such that the latter side of that person's ancestry would "dilute" by half the nobility he or she derived from the former side: (25)/2 = 32/2 = 16.
Some orders of chivalry limit their membership to persons who can prove a certain number of quarterings (e.g., sixteen for the Order of St. John).