I have a 10 gallon tank with two small goldfish in it, and was considering putting some (4-6) neon tetras in with them, for variety.

I've done some research on the tetras, and I've seen a lot of mixed opinions. Some say they are nippy, some say they don't bother anything. What's the truth??

And secondly, is adding more fish a no-no anyway?

first of all, fancy goldfish need 10 gallons each, and 20 gallons each for single tailed fish such as commons. Tropical fish with goldfish is not a good thing to do as thegoldfish will become very stressed which could eventually become fatal, and they might eat the tetras, plus the tetras need a different temp and different food

I had a common gf in with trops for awhile. He got on ok with the tetras although his tail had bites out of it now and then. He had more trouble with a molly and a swordtail who always wanted to chase him.

I had a goldfish tank at my parents' home and they were convinced by a petstore salesperson that they would be compatible.

My goldfish ate their fins.

I wouldn't try it.

Not only is it a bad idea. Goldfish have much different needs then a tetra as stated already your tank is to small for the two fish you have. So upgrade your tank for your goldfish and then get you a couple tetra's to go in the 10 gallon tank.

i have to bubbles with 4 tetras(little but not to little) and they are just fine. not sure if aussie fish are different than overseas fish.they get along good.

No they are not different then over seas fish and they still have different needs. Goldfish need cooler water then tetras. Goldfish need different food then tetras. Goldfish need huge amount of filtration and the current would be much for the tetras. So no they wont be fine for long.
I would highly recomend you getting a new tank for your tetras or for the goldfish lee.
Good luck