Can cherry barbs and neon tetras live together?

Yes, with caution.

Barbs categorized as "aggressive" is some what of a misnomer, when kept in group of under three they get bored and nip, which in turn would kill your neons, when kept in groups of 5 or more they keep to themselves.

I also have a school of the two neon's and rosy barbs and that i've got got here across that if my college of rosy barbs get somewhat low, they might develop right into somewhat aggressive with one yet another. If I even have around 8 or extra they are particularly calm. I even have not had any aggression from my neon's. I even have not encountered any disease from my neon's and a few of my neons are 3 years previous.

yes, just make sure the water quality is good and the tank has been established for months rather than just weeks. Neons can be difficult at first to keep alive.

I've kept these species together before with no problems.

Yes, you would need a tank of at least 30 gallons though.