with increased mesh refinement, your stable time decreases (proportionally). As other has said, localized deformation. Another issue you should keep in mind that the more number of increments are involved, the more truncation/roundoff error is accumulated. In terms of element formulation, extra fine mesh leads to shear deformation becoming more dominant than bending. the element size must satisfy the underlying assumptions that were taken in formulation. Lastly, when very fine mesh is used, it's always recommended to run explicit in double precision, and output_precision=full
The explicit method has to satisfy the CFL stability criterion (Courant-Friedrichs-Levy condition) which states that the time step should be so short that a wave-like disturbance cannot travel through the smallest finite element. You are on the safe side with the longitudinal wave (compressional wave) with speed cp=√(λ+2 μ)/ρ in the case of linear elasticity. The CFL criterion requires smaller time steps as the mesh is refined.
I think you got your answer. The step size depends on the element size.
We should be interested in the maximum step size, since the minimum is meaning less here. We are interested to get answer faster, thus the maximum step size interests us.