More stable is in my opinion not a very clear term with regards to your question. For the time integration of a semi-discrete system, less spatially refined discretisation tends to be more stable (see e.g. the largest eigenvalue of the discrete Laplacian which scales with the square of the number of points) when integrated with an explicit scheme, but this may not affect well chosen implicit schemes.
Regarding the error, I know that some differential-algebraic equations may produce a "hump" phenomena where the error temporarily increases for some variables as the time step is decreased. See e.g. Hairer & Wanner "Solving Ordinary Differential Equations II".
For ODEs producing a marginally unstable dynamics, it is also possible that there exists a threshold step size above which an implict time integration scheme stabilises the solution, thus the error would be relatively independent of the step size in that range.
For PDEs, it may also depend on how you measure the error. I often have seen that if I measure a local error(e.g. at one point in space), it may change sign at one point in the convergence study, thus actually crossing 0 for a specific discretisation, and later converging to 0 for finer discretisations. In that case, computing a global space-error or an average error on multiple points usually restores the correct monotonous behaviour.