The underlying concept of a “natural” monopoly is that costs for a given output are minimized when one firm produces that output. This paper shows that a paradox arises for certain infrastructure and delivery products satisfying widely accepted technical criteria for such monopolies: Under well-defined conditions, total costs can be reduced after the entry of a competitor. This finding is important if special state benefits, like protection against entry or direct subsidies, are provided to firms meeting the criteria. An alternative definition of natural monopolist is proposed that removes the counter-intuitive result. Plausible boundary conditions are derived for entry to reduce total costs. Surprisingly, applying the new definition to postal delivery, incumbents with low letter volumes least merit state protection, suggesting an increasing challenge from digitalization.