Engine cooling fan noise is a relevant issue for manufacturers. It is well known that both the operating point and testing environment can affect the noise generation mechanisms and, consequently, the measured noise may change. Therefore, these aspects are investigated on a reference industrial fan, for which experimental data exists, using high-fidelity numerical simulations based on the lattice-Boltzmann method. Two operating conditions, namely the free blowing and the maximum efficiency ones, and three testing environments are analyzed: (i) a conventional semi-anechoic room, (ii) an ideal free field environment, and (iii) a testing environment resembling an anechoic aeraulic facility. For cases (i) and (ii) no pressure difference across the fan is imposed, while, for case (iii), a pressure difference across the fan can be imposed. For the latter, the impact of a fully reflective and fully absorbing wall separating the two regions upstream and downstream of the fan is analyzed. At free blowing conditions, the flow over the blades is largely separated. When the blade passes through a blockage region, because of the presence of a honeycomb-like structure needed for structural purposes, it experiences a prominent loading hump. The far-field noise, at a listener located along the axis of rotation, is therefore highly tonal, with a clear peak at the blade passing frequency tone. When the same fan is tested in a free field environment, it is found that there is a difference in the acoustic pressure at higher harmonics of the blade passing frequency due to the presence of flow recirculations in the anechoic room. Placing a thin wall across the fan increases the mass flow rate, for a given rotational speed, which results in a more severe flow separation over the blades and, therefore a higher tone prominence at the blade passing frequency. If the thin wall is modeled as a sound-absorbing wall, there is a drop of the overall sound pressure level of about 2 dBA. When the fan is tested at its maximum efficiency, i.e., nonzero pressure difference across the fan, it is found that the blockage effect is less relevant. The main noise generation mechanism is the back-flow vortex induced by the pressure difference across the fan interacting with the blade tip leading edge.