Saw a game where people with 4-of-a-kind Badugis were winning half of the pot in a Dramaha game. Saw someone with 3 Aces win half the pot with what they called a 3-card Badugi.

The game itself was Dramaha, 5 cards to each player, with a draw after the turn, the pot being split between the best Omaha hand on the board and whatever the best 5 card hand was (other variants will have the 5 card hand just be a traditional poker hand, a 2-7 hand, a count-the-pips hand, or in this case the best Badugi hand).

A Badugi hand is a low hand with the added condition that each card in the hand must be a different suit. The best hand is something like Ah 2s 3c 4d. A four card Badugi is any hand that has one card of each suit. Among four card Badugis lowest hand wins. A hand like
Ac Tc 4d 5d 2s would be a three card Badugi, in this example Ac 2s 4d. Any four card Badugi beats a three card Badugi. The board plays no role in the Badugi hand — the best Badugi dealt to a player gets half the pot.

Edit— just saw that it’s high card Badugi, not just Badugi. In that case a four card Badugi still must have one card from each suit, but among Badugis with the same number of cards, highest hand wins. I should have realized that when you said someone got half the pot with four aces — that would be the absolute nut high card Badugi (obviously it would have one card from each suit).

So would any 4-card Badugi beat a 3-card with 3 Aces?

Yes. Any 4 card Badugi beats any 3 card Badugi. 2h 3s 4c 5d plus any other card would beat Ac Ad Ah Kh Kc, for example.

Probably it’s a moot point but a 3 card Badugi always beats a 2 card Badugi. So 2c 3h 4s 5s 6s, for example beats Ac Ah Kc Kh Qc.