we're really falling apart, aren't we brother?

Though the one instance where he could reasonably get promoted would be in the event that 1) Mustang has some sort of opening in his sector and 2) Mustang feels like being a dick.

“Hey Fullmetal, long time no see.” “Yeah yeah Al and I are in a rush. We wanted to leave Central an hour ago so hurry up and spit it out what’d you call me here for?” “Oh don’t worry it’s good news. Wonderful news. You’re really making your way in the world. See Lieutenant Colonel Davis transferred recently to Southern Command–” “Oh no.” “–Which means there’s a spot for Lieutenant Colonel open here.” “No no no you didn’t.” “Longer hours, more responsibilities, sure. But better pay, and you’d get in good with some high-ranking people, assuming you know how to brown-nose.” “I will literally kick your ass, Mustang.” “For what? Brightening you future?” “No really I will kick your ass if you’re being real here–” “Is that a challenge, Lieuuuutenant Colonel Elric?” “I swear to God Mustang if you a c t u a l l y  d i d…”

This frustrates Mustang for some time, at least until Mustang discovers he actually is in possession of a real threat against Ed: Promotion.

This confuses a lot of people, who overhear fights between Ed and Mustang that end with Mustang menacingly discussing the better healthcare benefits of Lieutenant Colonels. Or the nicer pay. Or the guaranteed apartment housing in Central. 

Most soldiers start theorizing this is some kind of back-hand threat. Something like “You know Elric, you’re under consideration for Lt. Colonel and would get all these nice things. But if you’re going to be difficult, maybe we’ll give these to someone else? Think of what you’d be missing out on.” 

This theory loses all of its credibility one afternoon when Mustang’s soldiers overhear a loud, heated argument between the two of them from inside Mustang’s office. It ends with the sound of Mustang slamming both his hands down on his desk, kicking his chair back, and shouting, “I SWEAR TO GOD FULLMETAL IF YOU DON’T SHUT UP I WILL PROMOTE YOU!”

direct orders of "you really really need to kill this guy" ignored: 1

Fullmetal Alchemist 2003 : episode 49