When I started with trail runners, I thought that they would be sturdy enough with enough ankle protection. But, after backpacking...off trail.. with a heavy pack (above 60lbs), I am convinced now. My ankles are pretty weak from high school sports injuries. When I wear boots, I tend to roll them a lot more. For some reason, I almost never roll them when in trail runners.
A couple years ago, I rolled my ankle really bad...playing softball. I tried walking around with my heavy leather boots that have really stiff sides for "ankle support". They didn't help. Plus the weight of the boot just really made my ankle hurt more. A real ankle brace with trail runners was the only solution that worked for walking around. There are reasons to wear boots. Ankle support isn't one of them...bring a real ankle brace.