The problem has been determined to derive from six scripts which control the patrolling patterns of certain guards. The scripts, which run even if you are not in the area, exhausts the internal allotment of identification numbers which are assigned to every object in the game (e.g. dropped arrows, weaponry). Once all the space is full, newly created objects disappear from the game world.
The bug will occur if you have Shivering Isles installed, whether or not you even access the content. It will rear its ugly head 50 to 120 hours within gameplay, depending upon your framerate (we're not sure if this is hours of play after installation of expansion or total play time). The Xbox 360 version, running at 30 frames per second, is expected to hit the bug at 150 hours.
The Unofficial Elder Scrolls Pages have chronicled the issue and possible solutions. PC users can enjoy user-made patches; Xbox 360 owners have a suggested, although annoying and short-term (and spoiler-laden) fix they can use. We've contacted Bethesda for an official statement.
[Thanks to everyone who sent this in -- and to those with anecdotal evidence, we're sorry for your loss.]