The Arctic climate system is very sensitive to external perturbations, which results in more rapid surface air temperature (SAT) changes in the Arctic compared to lower latitudes. This study aims at assessing the performance of global climate models from the phases 5 and 6 of the Coupled Model Intercomparison Project (CMIP5 and CMIP6) in reproducing the periods of Arctic warming and cooling known from the observations. Two experiments are considered for 9 CMIP5 models, and one experiment is used for the 108 ensemble members from 32 CMIP6 models. The ensemble mean of unforced Arctic amplification variability in the CMIP5 pre-industrial control experiment suggests a crucial role of external forcings in the present Arctic amplification. The ensemble mean of the same models in the historical experiment shows a century-long upward trend and clearly reproduces the present-day Arctic amplification. However, there is no consistent response of increasing Arctic amplification in the early twentieth century warming period during the 1920s–1940s in the CMIP5 models. This indicates the 1920s–1940s Arctic warming was unlikely to happen due to external forcings. Using the large CMIP6 ensemble in the historical experiment we show that the multi-model ensemble mean in the new generation of high-resolution CMIP6 models does not reproduce that warming either, thus posing questions how well the models capture internal climate variability and distinguish it from natural and anthropogenic forcings. This issue ultimately affects the reliability of future Arctic climate projections.