Deep learning methods have shown outstanding classification accuracy in medical imaging problems, which is largely attributed to the availability of large-scale datasets manually annotated with clean labels. However, given the high cost of such manual annotation, new medical imaging classification problems may need to rely on machine-generated noisy labels extracted from radiology reports. Indeed, many Chest X-Ray (CXR) classifiers have been modelled from datasets with noisy labels, but their training procedure is in general not robust to noisy-label samples, leading to sub-optimal models. Furthermore, CXR datasets are mostly multi-label, so current multi-class noisy-label learning methods cannot be easily adapted. In this paper, we propose a new method designed for noisy multi-label CXR learning, which detects and smoothly re-labels noisy samples from the dataset to be used in the training of common multi-label classifiers. The proposed method optimises a bag of multi-label descriptors (BoMD) to promote their similarity with the semantic descriptors produced by language models from multi-label image annotations. Our experiments on noisy multi-label training sets and clean testing sets show that our model has state-of-the-art accuracy and robustness in many CXR multi-label classification benchmarks, including a new benchmark that we propose to systematically assess noisy multi-label methods. Code is available at https://github.com/cyh-0/BoMD.