I was idly browsing through TeX.SE, and at one point attempted to favourite ("star") a question that had already been starred more than a hundred times. Nothing happened. I tapped (yeah, touchscreen) higher on the icon, at which point the expected result occurred.
Further investigation revealed that the star count being sandwiched up into the icon (as reported here) causes it to overlap the star's clickable box in the page DOM; and as the count is the later DOM element, it prevents the lower part of the star being clickable.
The blue boxes are the overlapped areas. Pretty bad, isn't it?
The culprit is a single CSS property
.favoritecount { margin-top: -4px; }
with the effect of smashing the number into the star. Make that less negative and the problem goes away.