In the EU, the complex relationship between copyright and fundamental rights is accommodated in inter alia the Copyright Directive, which uses an internal system of exhaustive exceptions and limitations to copyright. However, it remains unclear whether fundamental rights can serve as autonomous grounds for limiting a copyright. Recently, the German Federal Supreme Court referred preliminary questions to the CJEU in the so-called Afghanistan Papiere case in order for the CJEU to address this unclarity. To sketch the context in which these questions have to be answered, this article explores the tension between copyright and fundamental rights in the EU legal order and identifies the boundaries of the system of exhaustive exceptions and limitations. After describing the EU legislative framework, it offers a typology of the CJEU's case law, before moving onto the ECtHR's Ashby Donald judgment, in which the issue of balancing copyright and other fundamental rights was addressed. It follows that whereas the Copyright Directive and the CJEU do not seem to allow for an external role for fundamental rights, this exact role seems to be required by the ECtHR for certain types of cases. This difference is assessed in light of the relationship of the two courts, which is essentially characterised by a need to prevent divergence between them. It is argued that the pending preliminary questions can probably be answered using the internal system of the Copyright Directive, but that they clearly indicate the limitations and problems connected to the CJEU's current approach towards copyright and fundamental rights.