Q:

What does it mean if your confidence interval does not contain the 'true' value?

I have a sample of a stochastic process (blue) for which I have developed point estimates for each time step (red). I have then proceeded to bootstrap 95% confidence intervals for the point estimates (black), which is shown in the figure. Now as you can see, the CI doesn't always include the true value of the process. How do I interpret this? 
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Does this mean that my way of finding the CI might be faulty? Or is this a result that can be expected from working with real data? 

A:

I don't see a problem. By definition:

If confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will approximately match the confidence level.

That doesn't mean your confidence interval must contain the true value. In statistics, you never know what the true value is. We don't know with 100% certainty if the true value is inside the interval or not, but we know if we repeat the process 100 times, about 95 of those intervals should bracket the unknown true value.

