We examine the dynamical evolution of wave packets in a cubical billiard where three quantum numbers (𝑛𝑥, 𝑛𝑦, 𝑛𝑧) determine its energy spectrum and consequently its dynamical behaviour. We have constructed the wave packet in the cubical billiard and have observed its time evolution for various closed orbits. The closed orbits are possible for certain specific values of quantum numbers (𝑛𝑥, 𝑛𝑦, 𝑛𝑧) and initial momenta (𝑘𝑥, 𝑘𝑦, 𝑘𝑧). We observe that a cubical billiard exhibits degenerate energy levels and the path lengths of the closed orbits for these degenerate energy levels are identical. In spite of the identical path lengths, the shapes of the closed orbits for degenerate levels are different and depend upon angles 𝜃 and 𝜙 which we term as the sweep and the elevation angles, respectively. These degenerate levels owe their origin to the symmetries prevailing in the cubical billiard and these levels disappear completely or partially for a parallelepiped billiard as the symmetry breaks due to commensurate or incommensurate ratio of sides.