My understanding is that the CLT is not related to the number of predictors but to the degree of non-normality of the data. So for a dataset that is very nonnormal, a very large sample size may be needed, and if there is a mild departure from normality not that much of a sample size. There is no specific cutoff number for linear regression. While if there are many predictors, a bigger dataset would be needed, this is due to overfitting, not to CLT.
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