Confidence interval interpretation in linear regression when errors are not normally distributed
In case you might want to try another way to find confidence intervals, and in addition to the nice and rigourous answer by Peter, I would also consider a resampling method like bootstrapping as a robust way to find confidence intervals.
One key advantage is that it does not assume any kind of distribution, being a distribution-free method to find your coefficients estimates.

In the case of finding a confidence interval for a linear regression, the steps would be:


Draw n random samples (with replacement) from your dataset, where n is the bootstrap sample size
Fit a linear regression on the bootstrap sample from step 1
Repeat steps 1 &amp; 2 n_iters times, where n_iters will be the number of bootstrap samples and linear regressions made on them
Now that we have n_iters values for the linear regression coefficients, we can find the interval limits via the min, median and max percentiles (e.g. for a 95% CI: percentile 2.5, 50 and 97.5) to find the coefficient estimate together with the CI limits




Please mind the variability of the confidence intervals along the x-axis values, taking into account the sampling error of the coefficients estimates (good source of read: https://greenteapress.com/wp/think-stats-2e/)

The associated code of my example including the plot can be found here