Biologically-inspired spiking neural networks (SNNs) hold great promise to perform demanding tasks in an energy and area-efficient manner. Memristive devices organized in a crossbar array can be used to accelerate operations of artificial neural networks (ANNs) while circumventing limitations of traditional computing paradigms. Recent advances have led to the development of neuromorphic accelerators that employ phase-change memory (PCM) devices. We propose an approach to fully unravel the potential of such systems for SNNs by integrating entire layers, including synaptic weights as well as neuronal states, into crossbar arrays. However, the key challenges of such realizations originate from the intrinsic imperfections of the PCM devices that limit their effective precision. Thus, we investigated the impact of these limitations on the performance of SNNs and demonstrate that the synaptic weight and neuronal state realization using 4-bit precision provides a robust network performance. Moreover, we evaluated the scheme for a multi-layer SNN realized using an experimentally verified model of the PCM devices and achieved performance that is comparable to a floating-point 32-bit model.