Common deep neural networks (DNNs) for image classification have been shown to rely on shortcut opportunities (SO) in the form of predictive and easy-to-represent visual factors. This is known as shortcut learning and leads to impaired generalization. In this work, we show that common DNNs also suffer from shortcut learning when predicting only basic visual object factors of variation (FoV) such as shape, color, or texture. We argue that besides shortcut opportunities, generalization opportunities (GO) are also an inherent part of real-world vision data and arise from partial independence between predicted classes and FoVs. We also argue that it is necessary for DNNs to exploit GO to overcome shortcut learning. Our core contribution is to introduce the Diagnostic Vision Benchmark suite DiagViB-6, which includes datasets and metrics to study a network's shortcut vulnerability and generalization capability for six independent FoV. In particular, DiagViB-6 allows controlling the type and degree of SO and GO in a dataset. We benchmark a wide range of popular vision architectures and show that they can exploit GO only to a limited extent.