Does an impossible value on a Confidence Interval for a probability distribution contain information? Suppose I analyze a distribution and determine that a 95% CI for some measure is [-1, 1] (bearing in mind the actual value in this case cannot be < 0). And suppose another 95% CI for some other measure is [0,1].
Can I, based on the CI, conclude anything different for these two CIs? What if the first CI was [-100,1]?
My intuition says that, eg, the [-100,1] example i should expect a lot more values near zero than in the [0,1] example.
The question boils down to: Does a CI that contains 'impossible' values contain information above the tradition interpretation of that CI, that conducting after the experiment a lot of times, 95% pf the results will be in [0,1]?
 A: Confidence intervals are formed by the "inversion" of probability statements about pivotal quantities.  There are a few ways you can get a confidence interval that encompasses impossible values for the parameter it is estimating.  One common way this occurs is when you use an approximation to the true distribution of the pivotal quantity (e.g., an asymptotic approximation) to form the interval.  The approximating distribution may give a non-zero probability of values outside the true support of the pivotal quantity, so that when you "invert" the probability statement, you get impossible values for the parameter in the confidence interval.
So, what does this mean?  Obviously the impossible values are impossible, so you can (and should) reduce your confidence interval to only include the possible values of the parameter.  If you get a confidence interval $\text{CI}=[-1,1]$ for a parameter that is known to be non-negative, then you should certainly adjust that to the interval $\text{CI}=[0,1]$.  However, the very fact that you are getting a substantial amount of coverage of values that are impossible is a big flashing warning sign that the distribution approximation underlying our interval is not very good, and so the confidence interval is not trustworthy.  In practice, if you get a confidence interval like this, you should change your method to use an alternative interval that does not use an approximation for the distribution of the pivotal quantity.
A simple example of this is when you generate a confidence interval for the probability parameter in an IID Bernoulli model.  If you use the normal approximation to the sample proportion then you get a confidence interval that can give bounds that are outside the range $0 \leqslant \theta \leqslant 1$.  However, if you use the Wilson score interval then this always respects the possible range of the parameter.
