Neural Networks (NNs) are increasingly used in the last decade in several demanding applications, such as object detection and classification, autonomous driving, etc. Among different computing platforms for implementing NNs, FPGAs have multiple advantages due to design flexibility and high performance-to-watt ratio. Moreover, approximation techniques, such as quantization, have been introduced, which reduce the computational and storage requirements, thus enabling the integration of larger NNs into FPGA devices. On the other hand, FPGAs are sensitive to radiation-induced Single Event Upsets (SEUs). In this work, we perform an in-depth reliability analysis in an FPGA-based Binarized Fully Connected Neural Network (BNN) accelerator running a statistical fault injection campaign. The BNN benchmark has been produced by FINN, an open-source framework that provides an end-to-end flow from abstract level to design, making it easy to design customized FPGANN accelerators, while it also supports various approximation techniques. The campaign includes the injection of faults in the configuration memory of a state-of-the-art Xilinx Ultrascale+ FPGA running the BNN, as well an exhaustive fault injection in the user flip flops. We have analyzed the fault injection results characterizing the SEU vulnerability of the circuit per network layer, per clock cycle, and register. In general, the results show that the BNNs are inherently resilient to soft errors, since a low portion of SEUs in the configuration memory and the flip flops, cause system crashes or misclassification errors.