Hydrogen produced from renewable electricity can play an important role in deep decarbonization of industry, such as primary steel-making. However, adding large electrolyzer capacities to a low-carbon electricity system also increases the need for additional renewable electricity generation which will mostly come from variable renewable energies (VRE). This will require hydrogen production to be variable, unless sufficient flexibility is provided by other sources. Existing sources of flexibility in hydro-thermal systems are (a) hydropower and (b) thermal generation. However, increasing the flexibility of hydropower generation may have negative consequences for river ecosystems and the use of fossil and non-fossil fuels in generation may increase if thermal power is increasingly used to balance short-falls in wind power during electrolyzer operation. We assess here for our Swedish case study the utilization of electrolyzers with a dispatch model, assuming that additional VRE generation matches the additional electricity demand of hydrogen production on average. The flexibility of hydropower and thermal generation is restricted in four scenarios, and we run our model for 29 different weather years to test the impact of variable weather regimes. We show that (a) in all scenarios, electrolyzer utilization is above 60% on average, (b) the inter-annual variability of hydrogen production is very high if thermal power is not dispatched for electrolysis, (c) this problem is aggravated if hydropower flexibility is also restricted, and therefore (d) either long-term storage of hydrogen, backup hydrogen sources, or additional flexibility measures may be necessary to guarantee continuous hydrogen flows, and (e) adding wind power and electrolysis decreases the need for other backup flexibility measures in the system during climatic extreme events.