We claim that a source of severe failures for Variational Auto-Encoders is the choice of the distribution class used for the observation model. A first theoretical and experimental contribution of the paper is to establish that even in the large sample limit with arbitrarily powerful neural architectures and latent space, the VAE fails if the sharpness of the distribution class does not match the scale of the data. Our second claim is that the distribution sharpness must preferably be learned by the VAE (as opposed to, fixed and optimized offline): Autonomously adjusting this sharpness allows the VAE to dynamically control the trade-off between the optimization of the reconstruction loss and the latent compression. A second empirical contribution is to show how the control of this trade-off is instrumental in escaping poor local optima, akin a simulated annealing schedule. Both claims are backed upon experiments on artificial data, MNIST and CelebA, showing how sharpness learning addresses the notorious VAE blurriness issue.