In fact, both would be enough. Problem is, if they were all somehow attracted to or biased towards the mean (because samples are taken with either the covariate values, or the slope/intercept distributions being fixed to a point), then the samples will vary less than the model.
No, neither is the case. I am well able to produce any representation from my data. Goodness of fit is okay, except that the predictions always vary less than the actual data.
This might be due to shrinkage in my actual, hierarchical model. So to exclude that, I played with the test case in the file linked above. This is how the questions emerged.