Recently, speech enhancement (SE) based on deep speech prior has attracted much attention, such as the variational auto-encoder with non-negative matrix factorization (VAE-NMF) architecture. Compared to conventional approaches that represent clean speech by shallow models such as Gaussians with a low-rank covariance, the new approach employs deep generative models to represent the clean speech, which often provides a better prior. Despite the clear advantage in theory, we argue that deep priors must be used with much caution, since the likelihood produced by a deep generative model does not always coincide with the speech quality. We designed a comprehensive study on this issue and demonstrated that based on deep speech priors, a reasonable SE performance can be achieved, but the results might be suboptimal. A careful analysis showed that this problem is deeply rooted in the disharmony between the flexibility of deep generative models and the nature of the maximum-likelihood (ML) training.