Repeatability of confidence intervals of variable confidence I am aware that the information value of confidence intervals is still debated. However, I would like to keep the discussion to a Statistics 101 level.
Say we compare 99% CI and 95% CI. A 99% confidence level requires more trust than a 95% confidence, so how can you make the interval more trustworthy? Make it wider, of course. The 99% CI is wider than the 95% CI.
This "more trustworthy" phrase reminds a college student the previous chapter in the 101 textbook: something that is good at measuring the things it intends to measure is said to be accurate. Therefore, the wider 99% CI should be more accurate than the 95% CI.
Since there is a trade-off between precision and accuracy, it follows that the narrow 95% CI must be more precise than the 99% CI.
This makes sense for people who have been spared Statistics 101, because they have a different understanding of the word "precise". For them, narrow intervals are precise. But, for the statistics student, precision is described as the same as repeatability. So all the above suggests that the measurement /  calculation of a 95% CI more repeatable than that of a 99% CI.
This seems wrong: there should be no difference between the way the 99% CIs and the 95% CIs are distributed, when N is kept constant. Both types of CI are centered around sample means, which in turn are always on the normal distribution predicted by central limit theorem.
Are the calculations of 95% CIs more repeatable than those of 99% CIs? Where did I go wrong?
 A: Statistics 101 is a very difficult subject to teach, mostly because one has to introduce many things that have a different meaning in statistics and mathematics than in real life.
So, the shortest answer to the question is "you mixed something". A confidence level is not much more than a parameter that is used in a strictly defined procedure. If a confidence interval is determined for some data/experiment/model, then the boundaries will depend on the same data/input, as well as this parameter. So far the technicalities.
Now, if one turns to interpretation or even meaning, one has to be very careful in phrasing. The usual, slightly frequentistic interpretation is: if you would repeat the procedure under the same circumstances a large number of times, the "true" value would lie with the given probability of the confidence level inside the calculated confidence interval. However, this interpretation is usually more fictional than something real. The first thing to think about is repeatability; but maybe there is no "true" value, because of some other design defect. 
If one is more subjective in the interpretation of probabilities, then one would take the confidence level as a level of believe or believability. But even then you would have the factor in, that your assumption might be wrong.
If you leave the field of statistical terminology with "trustworthy" or "preciseness", then you'll find that you have no handle on this. Precision can be (and sometimes is) expressed as the inverse variance. But this variance is not accounted for or even described by a confidence interval.
To give an example for both, take the measurement of decays of certain instable particles. Example A is the free neutron, with lots of data and careful analysis one finds a result in the order of 15 minutes with some uncertainty. Example B is the proton, with even more data and much more careful analysis one finds that the lifetime of it has to be larger than $10^{31}$ years, so about 21 orders of magnitude longer than the age of the universe. The results are calculated very similar and for a given confidence level you'll get a certain value -- in the case of the proton only a lower boundary. But if either value is trustworthy is not a feature of the confidence level.
A: 
Say we compare 99% CI and 95% CI. A 99% confidence level requires more
  trust than a 95% confidence, so how can you make the interval more
  trustworthy? Make it wider, of course. The 99% CI is wider than the
  95% CI.
This "more trustworthy" phrase reminds a college student the previous
  chapter in the 101 textbook: something that is good at measuring the
  things it intends to measure is said to be accurate. Therefore, the
  wider 99% CI should be more accurate than the 95% CI.

Using a wider CI simply means that if you repeat the experiment on the same population with an independent random sampling using the same procedures as before you are more likely to get a result within the range of the CI. It is no more nor less accurate. Confidence intervals do not measure any aspect of accuracy, which is basically the difference between your estimate and the true value. The confusion may arise because we unfortunately often don't know the true value and are doing the experiment to get an estimate of it. But no amount of widening CIs will make up for not knowning the true value. What it does is make it more likely that you will 'succeed' in validating that estimate fits inside your huge limits.

Since there is a trade-off between precision and accuracy, it follows that the >narrow 95% CI must be more precise than the 99% CI.

This is an attempt at logical follow on, but as we saw above the premise was wrong and so the conclusion is inevitably wrong. However, precision and accuracy trade off is true in quantum mechanics it is not the case in any classical science where measurement does not significantly perturb the sample being analysed. In these cases accuracy and precision are orthogonal information and can be optimised independently. So even the follow on logic is flawed.
Furthermore, CIs do not measure precision directly, they measure precision adjusted for sample size. If you always know the sample size you can trace back to get precision, but you can't if you don't have that information to hand.

This makes sense for people who have been spared Statistics 101,
  because they have a different understanding of the word "precise". For
  them, narrow intervals are precise. But, for the statistics student,
  precision is described as the same as repeatability. So all the above
  suggests that the measurement / calculation of a 95% CI more
  repeatable than that of a 99% CI.
This seems wrong: there should be no difference between the way the
  99% CIs and the 95% CIs are distributed, when N is kept constant. Both
  types of CI are centered around sample means, which in turn are always
  on the normal distribution predicted by central limit theorem.

You are right, this is wrong because of the reasons described above.

Are the calculations of 95% CIs more repeatable than those of 99% CIs?

No, because 95% CIs and 99% CIs are based on the same precision and same N weighted by a difference z factor. The mistake stems from the very source you suspect, a misunderstanding of precision.
