I have managed to crash my frontline logistics in many ways so far:
Building too many assets at the same time (ironically usually logistics assets).
Too many armored/motorized units moving and attacking during the same turn.
Getting caught up in the attack mentality and not taking the time to make sure there are truck stations and supply bases to maintain things.
Raising large formations that suck up fuel and ammo to make them ready.
Probably using my level bombers that use a ridiculous amount of fuel and ammo (now that you pointed it out). Sadly I have not had the chance to try thopters or jets as of yet.
The need to stop and take a turn or two off during offensives to allow for your logistics to catch up to the action is crucial. I am, maybe, finally learning this!