Experiment to identify of all the poles from the input output data irrespective of pole zero cancellation?
Not if cancellation is perfect.

However, if there is a very slight difference between a pole and its cancelling zero, this produces a 'dipole' (control engineering term, not an antenna!); and a dipole shows up as a long tail on the step response (the step response takes a significant time to reach its steady state). The larger the difference, the larger the tail; the smaller the difference, the smaller, but more persistent is the tail.