# Does high number of values outside of 95% Confidence Interval imply non-normality?

I am looking at the errors of my model, i.e. difference between predicted outputs and actual values. Finding a mean and standard deviation, I found that many values (sometimes more than 50%) are outside of 95% confidence interval. Does this mean that my error is no normally distributed? Is that even possible? Shouldn't 95% CI mean 95% of my values be in this range?

Put another way, the confidence limits are about $\pm 2\sigma/\sqrt{n}$, while the 95% limits for a normal population are about $\pm 2\sigma$, without dividing by $\sqrt{n}$.