Quantum systems in mixed states can be unentangled and yet still nonclassically correlated. These correlations can be quantified by the quantum discord and might provide a resource for quantum information processing tasks. By precisely controlling the interaction of two ionic qubits with their environment, we investigate the capability of noise to generate discord. Firstly, we show that noise acting on only one quantum system can generate discord between two. States generated in this way are restricted in terms of the rank of their correlation matrix. Secondly, we show that classically correlated noise processes are capable of generating a much broader range of discordant states with correlation matrices of any rank. Our results show that noise processes prevalent in many physical systems can automatically generate nonclassical correlations and highlight fundamental differences between discord and entanglement.