Hydrogen-fired gas turbines have the potential to play an important role in decarbonized energy sectors. However, a demand-oriented supply of CO2-neutral hydrogen for the operation of gas turbines is technically and economically challenging. These challenges mainly arise due to various interdependencies between the volatility of renewable power generation, available hydrogen production capacities, available hydrogen storage capacities and the operational demands to be met by gas turbines. The present study aims to quantify these interdependencies by conducting a detailed model-based analysis of an exemplary combined heat and power system featuring a hydrogen-fired industrial gas turbine with on-site hydrogen production via electrolysis and on-site hydrogen storage in pressure vessels. In order to identify the sought-after interdependencies, simulation runs featuring various combinations of available hydrogen production and storage capacities are analyzed. If only local power surpluses are utilized for the electrolysis, the obtained results reveal a strong non-linear impact of both the hydrogen production capacity and the storage capacity on the ability to provide hydrogen for the gas turbine. Furthermore, the results indicate that an exclusive utilization of local power surpluses leads to very limited periods of hydrogen-based gas turbine operation and low utilization rates of the available hydrogen production and storage capacities. If additional power for the operation of electrolyzers is supplied by the grid, increased utilization rates and prolonged periods of hydrogen-based gas turbine operation can be achieved. However, in order to realize an overall reduction of CO2 emissions, this mode of operation requires the supply of large quantities of renewable power by the grid. Furthermore, the results of an additional economic analysis reveal that both investigated operational modes are currently not economically viable within the considered economic framework.