Increasing evidences show that the abnormal microRNA (miRNA) expression is related to a variety of complex human diseases. However, the current biological experiments to determine miRNA-disease associations are time consuming and expensive. Therefore, computational models to predict potential miRNA-disease associations are in urgent need. Though many miRNA-disease association prediction methods have been proposed, there is still a room to improve the prediction accuracy. In this paper, we propose a matrix completion model with bounded nuclear norm regularization to predict potential miRNA-disease associations, which is called BNNRMDA. BNNRMDA at first constructs a heterogeneous miRNA-disease network integrating the information of miRNA self-similarity, disease self-similarity, and the known miRNA-disease associations, which is represented by an adjacent matrix. Then, it models the miRNA-disease prediction as a relaxed matrix completion with error tolerance, value boundary and nuclear norm minimization. Finally it implements the alternating direction method to solve the matrix completion problem. BNNRMDA makes full use of available information of miRNAs and diseases, and can deals with the data containing noise. Compared with four state-of-the-art methods, the experimental results show BNNRMDA achieved the best performance in five-fold cross-validation and leave-one-out cross-validation. The case studies on two complex human diseases showed that 47 of the top 50 prediction results of BNNRMDA have been verified in the latest HMDD database.