With dc magnetization and polarized neutron reflectometry we studied the ferromagnetic response of YBa₂Cu₃O₇/La_2/3Ca_1/3MnO₃ (YBCO/LCMO) multilayers that are grown with pulsed laser deposition. We found that whereas for certain growth conditions (denoted as A type) the ferromagnetic moment of the LCMO layer is strongly dependent on the structural details of the YBCO layer on which it is deposited, for others (B type) the ferromagnetism of LCMO is much more robust. Both kinds of multilayers are of similar structural quality, but electron energy-loss spectroscopy studies with a scanning transmission electron microscope reveal an enhanced average Mn oxidation state of +3.5 for the A-type as opposed to the B-type samples, for which it is close to the nominal value of +3.33. The related, additional hole doping of the A-type LCMO layers, which likely originates from La and/or Mn vacancies, can explain their fragile ferromagnetic order, since it places them close to the boundary of the ferromagnetic order at which even weak perturbations can induce an antiferromagnetic or glassy state. On the other hand, we show that the B-type samples allow one to obtain YBCO/LCMO heterostructures with very thick YBCO layers and, yet, strongly ferromagnetic LCMO layers.