We have mapped from the quantum to the classical limit the spin excitation spectrum of the antiferromagnetic spin-1 Heisenberg chain system CsNiCl3 in its paramagnetic phase from T=5 to 200 K. Neutron scattering shows that the excitations are resonant and dispersive up to at least T=70 Ksimilar or equal to2.7J, but broaden considerably with increasing temperature. The dispersion flattens out with increasing temperature as the resonance energy Delta at the antiferromagnetic wave vector increases and the maximum in the dispersion decreases. The correlation length xi between T=12 and 50 K is in agreement with quantum Monte Carlo calculations for the spin-1 chain. xi is also consistent with the single mode approximation, suggesting that the excitations are short-lived single particle excitations. Below T=12 K where three-dimensional spin correlations are important, xi is shorter than predicted and the experiment is not consistent with the random phase approximation for coupled quantum chains. At T=200 K, the structure factor and second energy moment of the excitation spectrum are in excellent agreement with the high-temperature series expansion.