I got a second 5 gallon aquarium for my new betta. I would like to add another species and was thinking of neon tetras. Are they compatible with bettas? I've had problems with Serpae tetras before.

They CAN...but...Personally, for a 5 gal I wouldn't put anything else in with the betta except for something like some little Pygmy Corydoras cats (Corydoras Habrosus), shrimp, or a snail or two. Bettas don't like things that swim in their "space," and while neons and bettas normally can get along, I've seen them more prone to getting into fights with each other in anything smaller than 10 gals.

Small catfish, snails, and shrimp are good choices.

Though this will also depend on your betta's individual personality and whether they will ignore or attack. Some bettas get along with most everything, others will attack everything. My last betta was too aggressive to tolerate anything at all (he would even attack snails!), so he had to live by himself. He liked to attack my finger regularly. My current one is pretty mellow and does fine in community tank setting.

I have a bronze cory in my 10 gallon. It is nice and seems to be ignore and be ignored by all other fish most of the time, but I was shocked to see how big it can become.

Pretty much what the guy above me said. Maybe Pygmy cories as well, but 5 gallons is too small to consider putting other fish inside, try going for 10-20 gallons if you intend to have a betta + other fish tank.

I personally have 1 male betta in a 5 gal with 3 platies. He seems to ignore them for the most part. every now and then one of the platies will get to close and all the betta does is flare and the platie runs away. Also the betta is very active!!!

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20 Gallon, 6 zebra dinos, 4 green tigers, 3 dojo loach's, 2 blue gourami's, 2 GBR's, 2 bristle plecto's, 2 longfin rosybarbs, 1 syno cat, 1 leopard puffer
25 Gallon, 2 clown loach's, 1 yoyo loach, 1 veil angle, 1 royal plecto, 1 female krib.
36 Gallon bowfront, 1 Green Severum, 2 Auratus, 1 Venustus, 1 Acei, 1 Kenyi, 1 Duboisi, 1 Demasoni, 2 other africans I haven't identified, 2 Pictus Cat, 1 Rainbow shark, 3 tertas, 1 Firemouth cichlid

I think you have too much in such a small tank. I previously had a betta with 3 platies, 3 serpaes, an algae eater and a bronze cory. They were almost doing well except that the serpaes were trouble makers (among themselves and with the betta)

I have a betta in a 10
gal tank, and when a tiger barb got sick, I took it out of my 75gal and put it into the 10gal tank for treatment. I kept a watchful eye on the betta and I didn't notice any aggression or stress. Now that the tiger barb is getting better, it is being more active so that might change.

Are most tetras not fin nippers? I'd think they would bother the betta even if the betta didn't bother them.

I've had glass cats, and oto's in with my Betta (s), without any difficulties. It does depend on their personalities, though. I have had nice Betta that don't bother anyone, and "spicy" betta's, who don't like anything else in their tanks