Deep learning (DL) has present great diagnostic results in fault diagnosis field. However, the poor interpretability and noise robustness of DL-based methods are still the main factors limiting their wide application in industry. To address these issues, an interpretable wavelet packet kernel-constrained convolutional network (WPConvNet) is proposed for noise-robust fault diagnosis, which combines the feature extraction ability of wavelet bases and the learning ability of convolutional kernels together. First, the wavelet packet convolutional (WPConv) layer is proposed, and constraints are imposed to convolutional kernels, so that each convolution layer is a learnable discrete wavelet transform. Second, a soft threshold activation is proposed to reduce the noise component in feature maps, whose threshold is adaptively learned by estimating the standard deviation of noise. Third, we link the cascaded convolutional structure of convolutional neutral network (CNN) with wavelet packet decomposition and reconstruction using Mallat algorithm, which is interpretable in model architecture. Extensive experiments are carried out on two bearing fault datasets, and the results show that the proposed architecture outperforms other diagnosis models in terms of interpretability and noise robustness.