If you did the CI calc and the result was beyond a practical value, then that indicates a poor model.

We are likely exploiting the law of large numbers, which states something like,
"If we calculate a statistic about many samples from the same population, we know that the statistic we calculate will be distributed in a normal distribution, regardless of the distribution of the thing we are measuring."

Meaning, I can present you with a device which has a hidden probability distribution of outputs. If I ask you the variance, you can take some trials and calculate the variance over those trials. You repeat this.

The law of large numbers says that YOUR calculations of the variance will follow a normal distribution. It does not state anything about the underlying distribution that motivates your measurements. I.e. Your measurements of the variance will have a mean, and that mean will have a spread to the data which follows a normal distribution. The variance you are measuring does not necessarily follow a normal distribution, but your measurements of that variance do.

So if you got a result that is outside practical limits of the input, it could indicate that you have a small sample.
Or that your model is poor.

*note that the variance of your measurement of the sample variance is characterized by the F-distribution, which also approaches a normal distribution for large numbers of experiments.