The BICEP2 Collaboration reports a detection of primordial cosmic microwave background (CMB) B mode with a tensor-to-scalar ratio r = 0.20(-0.05)(+0.07) (68% C.L.). However, this result disagrees with the recent Planck limit r < 0.11 (95% C.L.) on constraining inflation models. In this Letter we consider an inflationary cosmology with a preceding nonsingular bounce, which gives rise to observable signatures on primordial perturbations. One interesting phenomenon is that both the primordial scalar and tensor modes can have a step feature on their power spectra, which nicely cancels the tensor excess power on the CMB temperature power spectrum. By performing a global analysis, we obtain the 68% C.L. constraints on the parameters of the model from the Planck+WP and BICEP2 data together: the jump scale log(10)(k(B)/Mpc(-1)) = -2.4 ± 0.2 and the spectrum amplitude ratio of bounce to inflation r(B) ≡ P(m)/A(s) = 0.71 ± 0.09. Our result reveals that the bounce inflation scenario can simultaneously explain the Planck and BICEP2 observations better than the standard cold dark matter model with a cosmological constant, and can be verified by future CMB polarization measurements.