System: Win 7 SP1, Chrome 34.0.1847.116 m
I've seen this problem on a few sites. In my experience, it's a minority of them, but I've run into it on
two three separate sites today: English.SE, Android.SE, and on this very Meta.SE while writing this bug report. It's a little different on each of them, and English.SE is the worst, so I'll start there.
When entering the tags, the 2 x 3 dropdown appears, but the page doesn't make extra room below for the new page element that wasn't displayed before, and so the 2nd row of the tags dropdown appears behind the page footer. A second scrollbar appears for the frame the main page content appears in, but attempting to use it to scroll down closes the tags dropdown, and since that screen element is no longer visible, the frame fits again and the secondary scrollbar disappears. The length of this second scrollbar indicates to me when there is a 2nd row, or whether there's just the 3 tags in the top row, and the very bottom of that top row is all that's hiding behind the page footer.
On Android.SE, all of the first row of the tags dropdown is visible, and just the top part of the 2nd row is visible. Often, this is enough to identify the proper tag and no scrolling is necessary, however the secondary frame scrollbar that appears, allows one to scroll that frame down without closing the tags dropbox, so there's no "loss of functionality" (inability to see the 2nd row at all) like there is in English.SE.
While posting this bug report, I found the issue is on Meta.SE also. It's behavior is still different from the other two. The first full row of tags is visible, but the second row gets cut off, and no additional frame scrollbar appears.