Mine stayed on fine, too. I wasn't concerned about them coming off. They're made of the same material as the tags the airlines put on (you know, the tags with the three-letter airport codes). We did one kind of silly thing, though. There were four of us, so we received eight DME tags, but we only checked 2 suitcases, both wheeled bags. Each bag had two luggage handles, so we used four tags total, two on each bag, one for each handle.
If you follow the instructions on the back of the tag, the tag will be extremely secure. The instuctions tell you to remove the backing from each end of the tag (but not from the middle), and to affix the two exposed adhesive sections to each other.
If you only remove the backing from one end, the tag can come off easily. If you remove all of the backing, you'll have a sticky mess around your handle.
Again, just follow the instructions, and you won't need any other "ideas, hints, or tricks."
Thank you for answering the question that puzzled me when I got my tags yesterday (why not take all the backing off to make it more secure?). Common sense isn't my strong point sometimes.