The Copernican system actually had more epicycles than the Ptolemaic one.
But yes, you could keep adding epicycles to the Ptolemaic system and change parameters to match the accuracy even of today's planetary movements.
But epicycles couldn't explain hyperbolic movements of objects that only fly into the solar system once.
Keep in mind that any ellipse can be modelled by 2 epicycles.
Ellipses were a tighter constraint on available epicycle models.
Kepler's model actually involved hyperbola for comets, which cannot be explained by epicycles.