In this paper, the influence of harmonic and bounded noise excitations on the chaotic motion of a double well Duffing oscillator possessing both homoclinic and heteroclinic orbits is investigated. The criteria for occurrence of transverse intersection on the surface of homoclinic and heteroclinic orbits are derived by Melnikov theory, and are complemented by numerical calculations which display the bifurcation surfaces and the fractality of the basins of attraction. The results imply that the threshold amplitude of bounded noise for the onset of chaos moves upwards as the noise intensity increases beyond a critical value, which is further verified by numerically calculating the top Lyapunov exponents of the original system. Then we come to the conclusion that larger noise intensity results in smaller possible chaotic domain in the parameter space. The influence of bounded noise on Poincaré maps of the system response is also discussed, which indicates that when the noise intensity is less than some critical value, larger noise intensity results in larger area which the map occupies in the phase plane.