Code completion is one of the main features of modern Integrated Development Environments (IDE). Its objective is to speed up code writing by predicting the next code token(s) the developer is likely to write. Research in this area has substantially bolstered the predictive performance of these techniques. However, the support to developers is still limited to the prediction of the next few tokens to type. In this work, we take a step further in this direction by presenting a large-scale empirical study aimed at exploring the capabilities of state-of-the-art deep learning (DL) models in supporting code completion at different granularity levels, including single tokens, one or multiple entire statements, up to entire code blocks (e.g., the iterated block of a for loop). To this aim, we train and test several adapted variants of the recently proposed RoBERTa model, and evaluate its predictions from several perspectives, including: (i) metrics usually adopted when assessing DL generative models (i.e., BLEU score and Levenshtein distance); (ii) the percentage of perfect predictions (i.e., the predicted code snippets that match those written by developers); and (iii) the “semantic” equivalence of the generated code as compared to the one written by developers. The achieved results show that BERT models represent a viable solution for code completion, with perfect predictions ranging from ~7%, obtained when asking the model to guess entire blocks, up to ~58%, reached in the simpler scenario of few tokens masked from the same code statement.