The functionality that distributed ledger technology provides, i.e., an immutable and fraud-resistant registry with validation and verification mechanisms, has traditionally been implemented with a trusted third party. Due to the distributed nature of ledger technology, there is a strong recent trend towards using ledgers to implement novel decentralized applications for a wide range of use cases, e.g., in the financial sector and sharing economy. While there can be several arguments for the use of a ledger, the key question is whether it can fully replace any single trusted party in the system as otherwise a (potentially simpler) solution can be built around the trusted party. In this paper, we introduce an abstract view on ledger use cases and present two fundamental criteria that must be met for any use case to be implemented using a ledger-based approach without having to rely on any particular party in the system. Moreover, we evaluate several ledger use cases that have recently received considerable attention according to these criteria, revealing that often participants need to trust each other despite using a distributed ledger. Consequently, the potential of using a ledger as a replacement for a trusted party is limited for these use cases.