I bought five neon tetras to go with the one I have in my tank and they have all died with in the first two days I have had them. My nitrite is 0, ammonia 0, and nitrate 40. I know that's a little high but would it really kill them that quickly? Before they died they became sluggish, lost color, and would not eat and usually died 1-5 hours later. I have three catfish in that same tank and I'm so afraid they will get it! I have had them for five years :(. Does this sound like neon tetra disease?

There's no diagnosis possible with freshly arrived fish like that - they went so fast. It could have been neon disease (which is not just a neon problem) but also a bacterial infection, etc.
If it was neon disease it should have been evident at the store. The discolouration would already have been showing for them to go that quickly.

It sounds like these fish were on their way out by the time they hit your tank. How did you acclimate them, and how did you transport them?

I got them from Petco, and I floated their bag in the tank until it was the same temperature.

You should swap in a cup of water from your tank to their bag every 15 minutes or so for an hour to get them used to your water parameters. While floating them in the bag helps with temperature, it also leaves them stressed without cover in dirty water, and does not account for pH, hardness, and other parameters. You don't have to drip acclimate, but some acclimation will help with sensitive fish such as neons. Also, if you get the option cardinal tetras are a little easier to keep, and slightly more colorful.

I do it differently, and never allow pet store water into my tanks. Neons should be tougher than cardinals, but most cardinals are wild-caught while neons are farmed, hence the higher mortality rates on what should be a tougher species. Neons have recently become delicate, with whatever is being done wrong to increase their production.
Floating in the bag to equalize temperature is good, but I then pour off the bag water through a net and put the fish directly from the net to the new tank. If the store or breeder is local, then they are used to same water as I have here, and if they have been shipped, drip accimation leaves them in contact with the ammonia in the bag too long.
Doing this radical looking acclimation technique, I have not had a fish die or even look distressed in ten years or so.
The only time you must drip is if you get a fish from extremely hard water, and your water is softer. That is a serious problem, if you haven't adjusted the water they will go into.

I forgot to mention that part. After moving a couple cups of water, I also pour the water out through a net. You never know what sort of diseases are in fish store water.