How to test for correlated errors in regression I understand that one assumption that must hold for regression is for there to be no correlation in the error structure.
Put another way:

The residuals should be impossible to predict above chance.

I'm wondering how you might test this. For any set of observations $X_i$, could you simply train a model on all other $X_{j \neq i}$ and predict its residuals, conditioned on its covariates $x_1 ... x_k$ in stepwise fashion?
I'm not sure if I've worded that properly. But what I mean is: Could you simply take a row of data, train a model on all the other data you have available, and try to predict the residuals for that row while adding one covariate in at a time? Seems like you would have a serious multiple comparisons problem on your hands..
In short, is there a way to test for the independence of the errors assumption in LR?

There are several issues.
First, the idea of refitting the model with each observation left out and predicting the left out point is very commonly done (but more for detecting influential points than correlation in residuals).  There are computer routines and short-cuts for doing this (similar, but not exactly what you describe).  If you want to learn more about this look for any of the following terms: studentized residuals, externally studentized residuals, deleted residuals, jackknife residuals.
Next,  There are many ways that residuals can be correlated.  You need to understand how you data was collected and the science behind the data to understand which may apply in each case.  There are tests for different possible ways that the residuals could be correlated, but generally we use information outside of the data to choose which tests to do (trying to test for every possibility will lead to meaningless results).  Also note that these failure to reject the null in these tests does not guarantee that there is no correlation, just that that test did not have enough power to detect the level of correlation in the current data.  Knowledge of the science and some simple diagnostic plots can often lead to better understanding than just the tests.
Some common things that lead to the lack of independence include serial correlation (correlation within the time order of when the data points were collected); clusters (instead of a simple random sample, small groups were selected (families, dorm rooms, cities, etc.) and then additional sampling within those groups, residuals within a group may be closer related to each other than to residuals outside of the groups); non-linear relationships (fitting a straight line relationship with one of the x-variables when the true relationship is curved).
