Does this violate the assumption of independence for regression A very basic question which I have never encountered a discussion of before. I am conducting bivariate logistic regression (although my question applies to linear models as well). I have 11,500 distinct cases that closed either successfully or not (that is the dependent variable is their closure status).
My predictor is how many days of experience their counselor had when the case closed. There are about 400 distinct counselors (they handle many cases) so there are about 400 distinct levels of the predictor. Does the regression assume that each predictor is independent of each other? And if it does how would you approach, what method would you use, to address this issue?
 A: Your question is about cluster-robust inference, and the short answer is that typically, this does not change your estimate of a parameter (such as $\beta$ is a linear regression or a logistic regression), but it will affect your standard errors. Typically, two assumptions that are commonly made are that standard errors are uncorrelated across observations, and that the variance of the error term is constant (this is called homoskedasticity).
In the case of uncorrelated errors that are different, the extension for linear regressions is to compute White standard errors.
In your case, the issue is that standard errors are indeed correlated across observations, but in a particular way: they are correlated across the distinct counselors. This is called clustered errors, and many methods exist to accommodate clustering. See this Stackoverflow post for some R packages that allow for clustering in logistic regressions.
Additionally, I highly suggest you take a look (at least at the intro and first few sections) of this excellent introduction to clustered errors.
Edit
In response to OPs comment of this post, then the only thing worth adding is how you want to approach your model. You could model each counselor as a separate fixed effect in your logistic regression, and as mentioned in a comment of OPs post, you can also model as random effects. But typically, if you have enough counselors, then just looking at days of experience should be fine. Intuitively, if you had few counselors, even if you had infinite cases per counselor, you'd run into problems. Say you had 2, one with more days of experience, but is a bad person and thus a worse counselor. Then no matter how many cases you observe for those 2 counselors, you wouldn't be fine. So in your case, you want case numbers to be large and for number of counselors to also be large.
