The microscopic origin of superconductivity in the high-transition-temperature (high-T(c)) copper oxides remains the subject of active inquiry; several of their electronic characteristics are well established as universal to all the known materials, forming the experimental foundation that all theories must address. The most fundamental of those characteristics, for both the copper oxides and other superconductors, is the dependence of the superconducting T(c) on the degree of electronic band filling. The recent report of superconductivity near 4 K in the layered sodium cobalt oxyhydrate, Na(0.35)CoO2*1.3H2O, is of interest owing to both its triangular cobalt-oxygen lattice and its generally analogous chemical and structural relationships to the copper oxide superconductors. Here we show that the superconducting T(c) of this compound displays the same kind of behaviour on chemical doping that is observed in the high-T(c) copper oxides. Specifically, the optimal superconducting T(c) occurs in a narrow range of sodium concentrations (and therefore electron concentrations) and decreases for both underdoped and overdoped materials, as observed in the phase diagram of the copper oxide superconductors. The analogy is not perfect, however, suggesting that Na(x)CoO2*1.3H2O, with its triangular lattice geometry and special magnetic characteristics, may provide insights into systems where coupled charge and spin dynamics play an essential role in leading to superconductivity.