This paper proposes a fault-tolerant indoor localization system that employs Recurrent Neural Networks (RNNs) for the localization task. A decision module is designed to detect failures and this is responsible for the allocation of RNNs that are suitable for each situation. As well as the fault-tolerant system, several architectures and models for RNNs are exploited in the system: Gated Recurrent Unit (GRU), Long Short-Term Memory (LSTM) and Simple RNN. The system uses as inputs a collection of Wi-Fi Received Signal Strength Indication (RSSI) signals, and the RNN classifies the position of an agent on the basis of this collection. A fault-tolerant mechanism has been designed to handle two types of failures: (i) momentary failure, and (ii) permanent failure. The results show that the RNNs are suitable for tackling the problem and that the whole system is reliable when employed for a series of failures.