I have never broken a handlebar, though I know someone who has. This person was particularly hard on bikes, and, in an era when everything was light-weight aluminum, his bars just couldn't take it. I remember him bending the aluminum and then--I still shudder when I think of it--turn them upside-down so that next time it would bend them back to the normal position. Perhaps this made him an early pioneer of riser bars--but only temporarily.
It should come as no surprise that after a few of these bending-turning over-bending back episodes, his bars failed and broke right near the stem clamp.
Well since that time, I've not heard of a failure (not even with him). Handlebars these days are very strong. If you're really tough on bikes (freeride or downhillers), then you can even get chromoly bars. In fact, you can pretty much pick your bar material and weight these days--all with the assurance that they are tested better, and are much stronger than yesteryear.
Well, undoubtedly, a bar with 31.8 mm diameter at the stem is going to be stronger than 25.4 (mountain bars) or 26.0 (road bars). Unfortunately, I can't put my computer, GPS, or any of my bike lights on that size of bar. And guess what (this may come as no surprise): I don't feel like my cockpit is any more stable with the larger bars. I'm sure the bars are stiffer, but I never had a problem with my bars before. In fact, I prefer to use carbon bars because they damp out vibration and make for a more comfortable ride--something large-diameter aluminum just doesn't do. Another problem I have with the size is that it is significantly larger than the normal grip diameter, so the bar has to taper radically which, again, leaves an area of the bar which doesn't work well with accessories (computers, lights, etc.).
There is, however, one good thing that has come of this change. As I stated above, road and mountain bars have different diameters--that is, until now. At least now, we are moving toward a standard size. This should make things cheaper for manufacturers and give people more options. (Like Moustache bars on a mountain bike.)
Like it or not, these industry-wide decisions are generally made without consulting me, so I'm certain 31.8 will continue to gain in popularity. Perhaps soon the computer and light manufacturers will catch on and come up with better mounting options soon. Let's hope.