Recent development in additive manufacturing (AM) technique has led to advances in fabricating foams with desired internal structures. By using AM technique, we fabricate random foams (regularity = 0.0–0.8) and Kelvin foams (regularity = 1.0) with similar relative densities. The effects of cell regularity on the mechanical properties and deformation mechanisms of Voronoi foams are investigated experimentally and numerically. A series of quasi-static and dynamic compressive tests are performed and the strain distributions are obtained by the digital imaging correlation method. Experimental results indicate that random foams have lower collapse stresses and plateau stresses than Kelvin foams. It is due to that random foams show random localized deformations and cell-wall tearing failures while Kelvin foams exhibit layered deformation bands and no tearing failure. To decouple the two deformation and failure mechanisms affecting the Voronoi-foam mechanical properties, further numerical investigations without introducing tearing failure are carried out and random foams are found to have lower collapse stresses but similar plateau stresses with Kelvin foams. Therefore, it can be concluded that random foams have randomly-distributed localized deformations and thus can always mitigate the collapse stress, but may lower the plateau stress compared with Kelvin foams because of their cell-wall tearing failures. In addition, the cell regularity (0.0–0.8) has no significant influence on the mechanical properties of foams based on all experimental and numerical data.