Is there a situation under which the distribution of p-values is skewed towards 1? I heard that: 


*

*Under the null hypothesis, the distribution of p-values is flat over $[0,1]$.

*Under the alternative hypothesis, the distribution of p-values is skewed towards $0$.
Is there a situation under which the distribution of p-values is skewed towards $1$?
 A: It is also possible to have the effect in practice, when the null hypothesis is true but not all the assumption for the test are given. For example, the classical (non Welch) t-Test assumes equal variance in both groups. In the case that both groups are equally sized a violation is usually not that bad, otherwise the null distribution gets skewed. 
If the smaller group has a higher variance than the larger one the null distribution is skewed towards 0 and if it has a smaller variance it is skewed towards 1.
Some R Code for experimentation:
p.vals <- vector("numeric", 1e5)
for (i in 1:1e5) {
  x <- rnorm(5, 0, 1)
  y <- rnorm(50, 0, 10)
  p.vals[i] <- t.test(x,y, var.equal = TRUE)$p.value
}
hist(p.vals)


The example shown is the case where the larger group has higher variance. Note that a skewing of the null distribution towards 1 indicated the test is too conservative so results in more Type II Errors and skewing towards 0 gives too many false positives (Type I error).
A: That can happen in a one-sided test when your "true" parameter is inside the region of the null hypothesis but not on the boundary. Consider the following example in Stata where the "true" parameter (in this case mean) is 1:
clear all

program define sim, rclass
    drop _all
    set obs 100
    gen x = rnormal(1,1)
    ttest x = 0.75
    return scalar p1 = r(p_l)
    ttest x = 1
    return scalar p2 = r(p_l)
end

simulate p1=r(p1) p2=r(p2) , reps(20000) : sim
simpplot p1 p2, scheme(s2color) ylabel(,angle(horizontal)) ///
legend(order( 2 "H0: {&mu} {&ge} .75" 3 "H0: {&mu} {&ge} 1"))


I like this representation of the $p$-values. It shows on the y-axis the difference between the empirical estimate of the Cumulative Distribution Function (CDF) and the theoretical (continuous standard uniform) distribution. On the x-axis is the nominal $p$-value. The logic behind this graph is that for $p$-values in a simulation study in which the null hypothesis is true, the empirical CDF is an empirical estimate of the $p$-value. The empirical CDF gives for each nominal $p$-value an estimate of the probability of drawing a sample which deviates at least as much from the null hypothesis as the current sample (i.e. has a nominal $p$-value less than or equal to the current nominal $p$-value) if the null hypothesis is true. So negative values on the y-axis means that the emprical estimates of the $p$-value are less than the nominal $p$-values and positive values on the y-axis say that the empirical estimates of the $p$-values are larger than the nominal $p$-values. So the blue points correspond to a cumulative density function which bluges below the diagonal line which one would expect for a continuous standard uniform distribution. The corresponding histogram is shown below:

