The high-T_c cuprate superconductors are close to antiferromagnetic order. Recent measurements of magnetic excitations have reported an intriguing similarity to the spin waves - magnons - of the antiferromagnetic insulating parent compounds, suggesting that magnons may survive in damped, broadened form throughout the phase diagram. Here we show by resonant inelastic X-ray scattering on Bi₂Sr₂CaCu₂O_8+δ (Bi-2212) that the analogy with spin waves is only partial. The magnon-like features collapse along the nodal direction in momentum space and exhibit a photon energy dependence markedly different from the Mott-insulating case. These observations can be naturally described by the continuum of charge and spin excitations of correlated electrons. The persistence of damped magnons could favour scenarios for superconductivity built from quasiparticles coupled to spin fluctuations. However, excitation spectra composed of particle-hole excitations suggest that superconductivity emerges from a coherent treatment of electronic spin and charge in the form of quasiparticles with very strong magnetic correlations.