Remaining useful life (RUL) prediction of rolling bearings plays a crucial role in ensuring safe operation and maintenance decisions for equipment. However, due to the influence of monitoring location and working conditions, traditional deep learning methods are challenging to extract multi-dimensional and multiscale degradation features, decreasing the accuracy of RUL prediction. At the same time, there are uncertainties, such as noise and model parameters, which makes it difficult for RUL’s point prediction to meet maintenance requirements. A framework for bearing RUL interval estimation based on a cascaded multi-scale convolutional neural network (CMS-CNN) module is proposed. Firstly, depthwise separable convolution (DSC) and dilated causal convolution (DCC) constitute the main framework of the CMS-CNN module in the form of a cascade to realize multi-dimensional degenerate feature extraction in space and time. The convolution operation with different dilation rates is introduced into the module to achieve multi-scale feature extraction, and the convolutional block attention module (CBAM) is embedded to adaptively assign the importance of features. In addition, a staged-optimized mean-variance two-branched interval estimation output network layer is constructed to quantify the uncertainty of bearing RUL prediction results. Finally, the method is verified with two rolling bearing datasets. Experimental results show that the proposed method not only has high RUL prediction accuracy, but also accurately gives the uncertainty interval of the prediction results, which is better than some advanced prediction methods.