In Praetoria it is very possible to miss some badges, since your contacts forget you once you leave. Specifically, the badges for choosing a Loyalist outcome in a Morality mission (Loyalist), Resistance outcome (Resistance Member), and completing 5 Moral Choice missions (Moral High Ground). The badge Avid Reader in the Praetorian tutorial can also be missed since there is presently no way to go back to the tutorial.
I don't believe there are currently any missable badges on Primal Earth (someone correct me if I'm wrong). Efficiency Expert can be failed (and it only gets awarded to the mission owner, so you can't team for it) and isn't available in Ouro, but since Pither is a 45-50 contact that's not a big deal. The hero and villain explore badges (as well as the blueside Isolator defeat badge) in the tutorial can be obtained during the Ouroboros intro arc from the Pilgrim, which can be flashbacked in the 45-50 range.
Of course, it goes without saying that the anniversary event badges can be missed, so make sure you grab the 7th one this May. Other event badges also used to be missable, but in recent years it has become possible to obtain the previous year's badges each year when the event rolls around (Winter Event, for instance, allows you to purchase previous badges with Candy Canes).
"You don't lose levels. You don't have equipment to wear out, repair, or lose, or that anyone can steal from you. About the only thing lighter than debt they could do is have an NPC walk by, point and laugh before you can go to the hospital or base." -Memphis_Bill
We will honor the past, and fight to the last, it will be a good way to die...
Cheers, thanks for the info 
You can also miss the Mayhem/Safeguard exploration badges... but you could join a lowbies team to pick them up later.
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Originally Posted by ShadowNate
;_; ?!?! What the heck is wrong with you, my god, I have never been so confused in my life!
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I knew there was something I was forgetting.
"You don't lose levels. You don't have equipment to wear out, repair, or lose, or that anyone can steal from you. About the only thing lighter than debt they could do is have an NPC walk by, point and laugh before you can go to the hospital or base." -Memphis_Bill
We will honor the past, and fight to the last, it will be a good way to die...
Is it possible to level up and pass a badge by? I know there is the Oroboros or whomever who can send me back to stories I missed, but is there anything else that I may miss along the way just not knowing it was there and never be able to go back and get it?