Labels noise refers to errors in training labels caused by cheap data annotation methods, such as web scraping or crowd-sourcing, which can be detrimental to the performance of supervised classifiers. Several methods have been proposed to counteract the effect of random label noise in supervised classification, and some studies have shown that BERT is already robust against high rates of randomly injected label noise. However, real label noise is not random; rather, it is often correlated with input features or other annotator-specific factors. In this paper, we evaluate BERT in the presence of two types of realistic label noise: feature-dependent label noise, and synthetic label noise from annotator disagreements. We show that the presence of these types of noise significantly degrades BERT classification performance. To improve robustness, we evaluate different types of ensembles and noise-cleaning methods and compare their effectiveness against label noise across different datasets.