Is there a scenario where Bayes update results in no belief update when the prior has nonzero probability mass everywhere? If yes, in what settings can this occur?

Let $X \sim  U(a, a+1)$ for some unknown $a$ which is either 0 or 1. Suppose your prior on $a$ is uniform. Then suppose you observe $x = 1$. You can see via a symmetry argument that your posterior should be the same as your prior, since this gives you no information about $a$.
