Hello. We have four neon tetras in a 5-gallon tank that we bought two days ago. There are no other fish in the tank. Three of the fish are fine. But the smallest one has shown signs of distress since we added it to the tank. At first, it stayed near the top of the tank breathing rapidly and not joining the school. The next day it seemed to join the school off and on, but periodically it crashes to the bottom of the tank and rests. Just when we think it's near death, it will suddenly pop back up and start swimming with the school. Other times it appears to be skittish and stay away from the group. I haven't witnessed the feedings so it's hard to say whether it's getting enough to eat. Could all this be signs of stress due to being in a new tank? We're debating whether to separate the fish in case it's sick, or to give it time to get used to the new environment. Thank you.
Have you had your water tested? Do you have a filter on the tank? Have you used water cionditioner?
In order for a fish tank to sustain healthy aquatic life it needs a massive population of good bacteria living in the filter which clean and process the water for you, keeping everything safe and fresh. In new tanks, these bacteria simply don't exist and you get a lot of highly toxic "invisible" waste building up in the form of clear chemicals called ammonia and nitrite. These chemicals come when the fish poo, wee and uneaten food decomposes in the tank.
In a 2 day old, 5 gallon tank with 4 neons I can pretty much promise that your ammonia at least is at very dangerous levels. When ammonia gets high, it burns the fish, stops them getting enough oxygen (by burning their gills) and causes a lot of internal damage. The fish can look fins and then suddenly die, maybe weeks after first being exposed to the chemicals. Other fish are too small or fragile to cope and will get sick very quickly.
In order to keep healthy fish, you need to do one of two things: return the fish to the shop and perform a "fishless cycle" which grows those very special bacteria without harming fish, OR keep the fish but test the water 2 or 3 times a day for ammonia and nitrites and do large water changes to keep the chemicals at undetectable levels. Any detectable level will harm the fish.
Both methods require you to have test kits fot ammonia and nitrite which can be purchased online or at a good local fish shop (LFS).
Without doing these things, the fish will die a very, very painful death. Maybe not today, maybe not for a few weeks but eventually the poisons they have been living in will catch up with them.
Beginners' Resource Centre
That link contains almost everything you need to get started and keep your fish happy and healthy. Fish keeping is a very technical hobby and requires a lot of understanding and fine tuning. Please have a read, especially the sections on cycling and setting up and maintaining a new aquarium.
In a more longterm sense, neon tetras are not suited to such a small tank. They get to over an inch long and are very nervous, active fish that require a lot of swimming space, large groups (6 is a minimum) and lots of plants to hide in. They are also one of the least suitable and most miss-sold small fish for a new tank.
I would advise researching tiny fish (such as green neon tetra - different to neon tetra - and micro rasboras) or consider getting a betta (siamese fighting fish) instead.