We demonstrate the first downgrade attacks against RPKI. The key design property in RPKI that allows our attacks is the tradeoff between connectivity and security: when networks cannot retrieve RPKI information from publication points, they make routing decisions in BGP without validating RPKI. We exploit this tradeoff to develop attacks that prevent the retrieval of the RPKI objects from the public repositories, thereby disabling RPKI validation and exposing the RPKI-protected networks to prefix hijack attacks. We demonstrate experimentally that at least 47% of the public repositories are vulnerable against a specific version of our attacks, a rate-limiting off-path downgrade attack. We also show that all the current RPKI relying party implementations are vulnerable to attacks by a malicious publication point. This translates to 20.4% of the IPv4 address space. We provide recommendations for preventing our downgrade attacks. However, resolving the fundamental problem is not straightforward: if the relying parties prefer security over connectivity and insist on RPKI validation when ROAs cannot be retrieved, the victim AS may become disconnected from many more networks than just the one that the adversary wishes to hijack. Our work shows that the publication points are a critical infrastructure for Internet connectivity and security. Our main recommendation is therefore that the publication points should be hosted on robust platforms guaranteeing a high degree of connectivity.