I've noticed that Pine gets along quite well with one of my cats. He isn't quite bonded to me yet, we're at that kind of mutual stage where he isn't afraid of me but isn't quite in love. Anyways, recently I started letting him meet one of my cats, meep, a 3 year old female cat that still acts like a kitten. However, today I let him out of his cage and about an hour after play time, he went under my bed, and started barking. It surprised me so I assumed that he was barking for my cat since underneath my bed is where they always seemed to interact. After I brought my cat in the room, the barking stopped. I'm, of course, not going to leave them both alone in a room, but I was wondering if it was possible for sugar gliders to form bonds with predatory animals, such as cats?

Depends on the animal. Supervision is a must and I'd feed the cat before playtime with the glider just to be a Lil more safe. Mine play with my husky and my pug but our cat would not as she's a stalker and I do not trust her at all but. I've read/heard that many domestic animals don't bother the gliders as they are much different than say a mouse there scent can make housepets more curious than hostile.
Just be vigil and ready to separate them if need be. You gotta use your own discretion.

Be really careful when it comes to cats, you never know how they will react.
Gliders do not smell like mices, however, they do have prey movement, and the hunter instinct can always kick in (besides, in australia, cats are a plague for most of native marsupials, so, I'd be really careful with the idea that they don't considere gliders as prey because of the smell).
Even without hunting instinct, the cat may just want to play like they do with us, and a scratch or bite can turn into a fatal accident for a glider.

I know Tveit used to be bonded to a cat. He was never let out of the cage at his old home, but they had two cats. One would sleep on top of his cage. If Tveit woke up, he would bark until the cat came. And she seemed to know he was calling for her since the cat actually came to his cage. He would play with the cat's fur while she was sleeping on top of his cage.

It was really cute, but they never let him out of the cage with the cat, so I don't know if the cat would've attacked him or not. It's up to you to know your cat, just always be very careful if you decide to let them play together because the cat can be friendly normally and then if the glider does a movement that the cat views as prey, it will attack even if it doesn't normally.