The recently discovered Fickian yet non-Gaussian diffusion (FnGD) is here finely tuned and investigated over a wide range of probabilities and timescales using a quasi-2D suspension of colloidal beads under the action of a static and spatially random optical force field. This experimental model allows one to demonstrate that a "rapid" FnGD regime with a diffusivity close to that of free suspension can originate from earlier subdiffusion. We show that these two regimes are strictly tangled: as subdiffusion deepens upon increasing the optical force, deviations from Gaussianity in the FnGD regime become larger and more persistent in time. In addition, the distinctive exponential tails of FnGD are quickly built up in the subdiffusive regime. Our results shed new light on previous experimental observations and suggest that FnGD may generally be a memory effect of earlier subdiffusive processes.