Accurate indoor positioning for wireless communication systems represents an important step towards enhanced reliability and security, which are crucial aspects for realizing Industry 4.0. In this context, this paper presents an investigation on the real-world indoor positioning performance that can be obtained using a deep learning (DL)-based technique. For obtaining experimental data, we collect power measurements associated with reference positions using a wireless sensor network in an indoor scenario. The DL-based positioning scheme is modeled as a supervised learning problem, where the function that describes the relation between measured signal power values and their corresponding transmitter coordinates is approximated. We compare the DL approach to two different schemes with varying degrees of online computational complexity. Namely, maximum likelihood estimation and proximity. Furthermore, we provide a performance comparison of DL positioning trained with data generated exclusively based on a statistical path loss model and tested with experimental data.