In principle, a ‘natural' monopoly exists when a single firm has the lowest cost structure for production. This paper shows that a paradox arises for certain infrastructure and delivery products that satisfy a widely accepted technical definition for such monopolies. Under well-specified conditions, total costs can be reduced after the entry of a competitor. This finding matters because special state benefits, such as 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. These conditions suggest: (i) In infrastructure sectors, new investments should not be allocated automatically to natural monopolist incumbents; (ii) In postal delivery, incumbents with low letter volumes least merit state protection. Welfare implications are discussed.