Advanced numerical models used for climate prediction are known to exhibit biases in their simulated climate response to variable concentrations of the atmospheric greenhouse gases and aerosols that force a non-uniform, in space and time, secular global warming. We argue here that these biases can be particularly pronounced due to misrepresentation, in these models, of the multidecadal internal climate variability characterized by large-scale, hemispheric-to-global patterns. This point is illustrated through the development and analysis of a prototype climate model comprised of two damped linear oscillators, which mimic interannual and multidecadal internal climate dynamics and are set into motion via a combination of stochastic driving, representing weather noise, and deterministic external forcing inducing a secular climate change. The model time series are paired with pre-specified patterns in the physical space and form, conceptually, a spatially extended time series of the zonal-mean near-surface temperature, which is further contaminated by a spatiotemporal noise simulating the rest of climate variability. The choices of patterns and model parameters were informed by observations and climate-model simulations of the 20th century near-surface air temperature. Our main finding is that the intensity and spatial patterns of the internal multidecadal variability associated with the slow-oscillator model component greatly affect (i) the ability of modern pattern-recognition/fingerprinting methods to isolate the forced response of the climate system in the 20th century ensemble simulations and (ii) climate-system predictability, especially decadal predictability, as well as the estimates of this predictability using climate models in which the internal multidecadal variability is underestimated or otherwise misrepresented.