I know this is an old post, but I figured it out this past January and thought it was a unique enough problem that it deserved to be reported.
I have smoked taillights so I have Red LEDs for the rear blinkers/brake lights. It turns out that if one of the lights shorts on itself, the entire system goes haywire. So, when my headlights were on (taillights on as well) the system wouldn't blink. I figured this out sometime last fall. Annoying, but I made it work.
So, when I finally had a good amount of free time (on semester break) a buddy of mine and I took a shot at figuring out what the problem was. After 10 HOURS of looking at every flippin wire in the car, we finally found the culprit: it was one of the LEDs that wasn't working VERY WELL. I have the 12 LED bulbs, and about half of them were lighting up, so I didn't think anything of it. WRONG. After we'd exhausted all options, my buddy and I ended up playing "process of elimination" with the light bulbs and figured out that this SOB with about 5 WORKING LEDs was the one that was shorting the entire system.
Bought completely new LED bulbs and now the problem isn't a problem anymore. Stay thirsty my friends.