Should a two-sided binomial confidence interval for an observed frequency of zero have a right-tail probability of $\alpha$ or $\alpha / 2$? Suppose you have n Bernoulli trials and zero observed successes. The lower bound on a $1-\alpha$ two-sided confidence interval for the proportion of successes will of course be zero. By the definition of the Clopper-Pearson interval, the upper bound is $\{ \theta | P[Bin(n, \theta) \geq 0] \geq \alpha/2 \}$. 
My question is: isn't the coverage of this interval $1-\alpha/2$ instead of $1-\alpha$? 
Note: In R, binom::binom.confint returns an upper bound such that $ (1 - UB)^n = \alpha/2 $ when a $1 - \alpha$ confidence level is specified.
 A: Any frequentist confidence interval based on a normal approximation may give puzzling results when there are $0$ successes in $n$ trials.
It might be better to get a Bayesian probability interval estimate (sometimes called 'credible interval') based on a suitable beta
prior distribution. [The Bayesian approach to inference treats the success probability $p$ as a random variable--with a prior distribution to start, and finally a posterior distribution to reflect the prior distribution and the data.]
If you have little or no prior information,
you might use $\mathsf{Unif}(0,1) \equiv \mathsf{Beta}(1,1)$ as the prior. An alternative would be to use the Jeffreys prior $\mathsf{Beta}(.5, .5).$ 
If getting no successes does not come as a surprise, then
perhaps you want to use a prior distribution on the success
probability that puts more probability toward the lower end
of $(0,1).$
Suppose you use a uniform prior and get $x = 0$ successes
in $n = 20$ trials. Then the posterior distribution is
$\mathsf{Beta}(1+x, 1+n-x) = \mathsf{Beta}(1,21).$ Then
a one-sided 95% Bayesian interval estimate $(0, 0.0325)$ can be found in R as shown below. This interval suggests that the small success probability is likely below $0.0325.$
qbeta(.5, 1, 21)
[1] 0.03246822

If you really want a two-sided interval estimate for the binomial success probability, then you could use $(0.0012, 0.1611).$ [The Jeffries 95% two-sided interval is
$(0.000024, 0.11664).]$
qbeta(c(.025, .975), 1, 21)    # uniform prior
[1] 0.001204883 0.161097615

qbeta(c(.025, .975), .5, 20.5) # Jeffries prior
[1] 2.424648e-05 1.166390e-01

This interval makes it clear that you are not suggesting that
the events in question are completely impossible. For example,
if the task is to estimate the proportion of cattle in Montana
that have a particular disease, you are not saying that negative
results from 20 randomly chosen Montana animals should be taken as an indication that cattle throughout
Montana are totally free of the disease.
There are important philosophical differences between Bayesian
and frequentist statisticians, and these differences extend to
how interval estimates are interpreted by both groups of statisticians. However, frequentist statisticians are often
happy to use Bayesian computational methods for getting an interval estimate, and then calling the result a 'confidence interval' to be given a frequentist interpretation.
Note: For an extensive list of types of frequentist CIs for
the binomial success probability see Wikipedia, where assumptions are discussed. Notice that the Jeffreys interval, interpreted as frequentist, is among the first types of CIs discussed there.
