Understanding how feature learning affects generalization is among the foremost goals of modern deep learning theory. Here, we use the replica method from the statistical mechanics of disordered systems to study how the ability to learn representations affects the generalization performance of a simple class of models: deep Bayesian linear neural networks trained on unstructured Gaussian data. By comparing deep random feature models to deep networks in which all layers are trained, we provide a detailed characterization of the interplay between width, depth, data density, and prior mismatch. Random feature models can have particular widths that are optimal for generalization at a given data density, while making neural networks as wide or as narrow as possible is always optimal. Moreover, we show that the leading-order correction to the kernel-limit learning curve cannot distinguish between random feature models and deep networks in which all layers are trained. Taken together, our findings begin to elucidate how architectural details affect generalization performance in this simple class of deep regression models.