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Zero-shot neural machine translation is an attractive goal because of the high cost of obtaining data and building translation systems for new translation directions. However, previous papers have reported mixed success in zero-shot translation. It is hard to predict in which settings it will be effective, and what limits performance compared to a fully supervised system. In this paper, we investigate zero-shot performance of a multilingual EN$\leftrightarrow${FR,CS,DE,FI} system trained on WMT data. We find that zero-shot performance is highly unstable and can vary by more than 6 BLEU between training runs, making it difficult to reliably track improvements. We observe a bias towards copying the source in zero-shot translation, and investigate how the choice of subword segmentation affects this bias. We find that language-specific subword segmentation results in less subword copying at training time, and leads to better zero-shot performance compared to jointly trained segmentation. A recent trend in multilingual models is to not train on parallel data between all language pairs, but have a single bridge language, e.g. English. We find that this negatively affects zero-shot translation and leads to a failure mode where the model ignores the language tag and instead produces English output in zero-shot directions. We show that this bias towards English can be effectively reduced with even a small amount of parallel data in some of the non-English pairs.