We probe d-band quantum-well (QW) states in Pd nanofilms grown on Cu(001) using first- principles density functional theory (DFT) calculations combined with scanning tunneling spectroscopy (STS) experiments. This study reveals that QW states occur in the Pd overlayer films over a strikingly large film thickness and in a large binding energy range. The magnetic quantum number resolved orbital characters of these states are identified unambiguously by our DFT calculations. Calculations also demonstrate oscillatory multilayer relaxation and d-derived quantum size oscillation of the electronic density of states at the Fermi energy. The pseudomorphic growth, well-defined interface, and STS with single-layer thickness resolution allow us to probe individual QW states arising from the electron confinement along the growth axis of Pd films and to extract an accurate dispersion of the (∆₅-type) d electronic band, as these states are laterally highly localized and give rise to distinct and sharp peaks in the tunneling spectra.