Redundancy in a safety-critical system has the potential for greatly improving safety. However, in physical real life that potential cannot be realized if failure of an element of one subsystem can sometimes be physically related to failure of an element of another subsystem intended to be redundant with respect to the first. Two real-life element failures can be related to each other either because (1) one failure cascaded to cause the other, or (2) an abnormal event external to both elements caused both elements to fail. System designers and safety analysts should therefore be aware of all three types of real-life failure pairs: (1) unrelated, (2) cascading/consequential related, and (3) common-external-cause related. It is the possibility of occurrence of the latter two types of real-life failure pairs which is responsible for probabilistic dependencies between failures in probabilistic safety analyses. This paper clarifies the concepts of unrelated, related (both cascading/consequential and common-external-cause), independent, and dependent, failure pairs, their connections and their differences. It shows how the possibility of occurrence of a related failure pair in a real-life system results in a dependent failure pair in an analysis, and why and how much system safety can be degraded as a result. Methods are presented to help assess the degree of safety degradation which the possibility of related failure pairs can introduce into a system, so that their impacts can be reduced or even eliminated. The Event-Sequence Analysis method is used to illustrate the principles discussed. The paper finishes up with some observations and cautions about using fault-tree analysis to assess the safety of systems in which related failure pairs can occur.