Ensemble methods are a popular choice for learning from evolving data streams. This popularity is due to (i) the ability to simulate simple, yet, successful ensemble learning strategies, such as bagging and random forests; (ii) the possibility of incorporating drift detection and recovery in conjunction to the ensemble algorithm; (iii) the availability of efficient incremental base learners, such as Hoeffding Trees. In this work, we introduce the Streaming Random Patches (SRP) algorithm, an ensemble method specially adapted to stream classification which combines random subspaces and online bagging. We provide theoretical insights and empirical results illustrating different aspects of SRP. In particular, we explain how the widely adopted incremental Hoeffding trees are not, in fact, unstable learners, unlike their batch counterparts, and how this fact significantly influences ensemble methods design and performance. We compare SRP against state-of-the-art ensemble variants for streaming data in a multitude of datasets. The results show how SRP produce a high predictive performance for both real and synthetic datasets. Besides, we analyze the diversity over time and the average tree depth, which provides insights on the differences between local subspace randomization (as in random forest) and global subspace randomization (as in random subspaces).