Okay, it sounds like she has a bacterial infection. There's no way she could be pregnant because neons are egglayers, in fact that goes on the list of some of the worst advice I've ever heard from pet stores. Really bad advice like this usually goes with incorrect advice about stocking (ie telling you your tank can hold about three times the number of fish it can actually support) and no (or a very cursory) mention of cycling. Did they tell you anything about cycling?
It sounds like she's in a breeding tank at the moment. How big is it, does it have a filter and what other fish are in it? (are the other tetras neons and how many are there?) How big is her normal tank and what fish are in there? How long have these tanks been set up and are any other fish showing symptoms?
I'm going to go over the basics of cycling, there is more information in the pinned topics in the 'New To The Hobby' section. Basically, the job of your filter is not just to act as a 'sieve' and strain out all the muck in the water. Its real purpose is as a home for a big pile of bacteria that do all the work for you. Fish breathe out ammonia, they excrete ammonia as urine, and any waste in the tank (ie fish poo or uneaten food) breaks down quite quickly into ammonia. Ammonia is a deadly poison that kills fish. The purpose of the bacteria in the filter is to break down ammonia. One species of bacteria turns ammonia into nitrite (which is almost as bad as ammonia) and then another species turns nitrite into nitrate. This process is called the nitrogen cycle. A tank that has a healthy population of bacteria capable of processing all of the ammonia produced is said to be 'cycled'. In a cycled tank, there should never be a detectable level of ammonia or nitrite. A tank that is not cycled is, in short, a death trap - the ammonia will rise and rise until it kills all your fish, and the only way to get rid of it until the bacteria are established is to do water changes. Fanatically. Quite possibly several times a day, depending on how many fish are in the tank. Once the bacteria develop (this could take a few weeks) the water changes can go back to once a week or once a fortnight.
There are a lot of myths about cycling, most of them spread by pet stores. I'm assuming yours is not among the ones that hand out good advice about cycling, because they a) told you that a neon was pregnant (which is impossible) and b) told you to set up a breeding tank (which is pointless because neons are among the hardest fish to breed at home.)
- Various substances in bottles that claim to be bacteria DO NOT cycle your tank. Between the members on this forum, we've tried 'em all. Bio-Spira (USA), Bactinettes (UK) and Aquarium Science Eco-start (Aus) seem to work. They will not cycle a tank overnight, but they will reduce the time for the bacteria to establish. Bio-Spira and Bactinettes are refrigerated, Eco-start is freeze dried. Bacteria in suspension (little bottles of liquid) won't actually survive in a bottle on a dusty shelf - they are dead, and do nothing.
- Letting the tank run for a few weeks with no fish in it does NOT cycle it. The bacteria need time and a food source to become established. They will not grow in an empty tank with no fish.
- Adding one fish, letting it run for a few weeks, and then adding the rest of them doesn't work either.
What you're going through now is called a fish-in cycle - it means you're trying to establish your bacteria around the fish. What you need to do is to get a test kit - ASAP. I recommend the API Freshwater Liquid Master. DO NOT buy test strips, they are rubbish. The aim of a fish-in cycle is to keep the levels of ammonia and nitrite always under 0.25 parts per million by doing water changes. Higher than this, and you will permanently harm your fish. Once you tell us how many fish are in the tank we'll be better able to advise. If it's really overstocked you might have to return some to the store while you cycle because it's just so hard to keep the levels low enough when you have loads of fish in the tank.
As for the sick fish, a photo would be really good. Keep her in the 'breeding' tank - it's great that you have a second tank, because if you have to medicate her (which is quite likely) it will make it a lot easier. It sounds bacterial, but I can't really be sure. Is there any redness around the area? Also, did this start with an injury (like a bump or scrape) or just appear? Being extremely fat could be dropsy (which is usually caused by poor water quality, ie. the ammonia which is probably in your tank.) I honestly wouldn't expect a neon with dropsy to survive three days though. If you look at her from above, do her scales stick out and make her look like a pinecone?
/ WHAT WE NEED TO KNOW
Size of the breeding tank, what fish, does it have a filter?
Size of the main tank, what fish?
Are either of the tanks cycled?
Do you have a test kit, if yes, what are the test results for ammonia, nitrite, nitrate and pH? (if you don't have these it's ok, just get a test kit ASAP)
Does the sick fish look like a pinecone from above? Is the infected area red? Are any of the other fish showing any symptoms of illness?