Like brecon, I too learned the hard way about the need for a quarantine tank. But even with that, I have on occassions moved the fish into the main tank too quickly and introduced disease into the main tank. I have a plain basic 25 litre tank as my quarantine tank, though at the moment it has my baby cherry shrimp in.
Twice over the last 17 years I have got up in a morning to find every fish lying dead on the bottom. That's every single fish, nothing left alive. The second time was 2003 (it was the day the men came to fit the double glazing, that's how I remember) which was before I used the internet. I hadn't a clue on either occassion what had happened, though I have my suspicions now - and it was my fault. Each time, I removed the dead fish, did a water change and got more fish.
I was pretty lazy back then about water changes, sometimes I'd go a month between them, and I was also overstocked. Lots of fish make lots of ammonia which the filter bacteria turn into lots of nitrate. Nitrate is acidic and if the water has very little carbonate it soon gets used up leaving nothing to buffer the water against pH changes. I now know that I have very little carbonate in my tapwater. I strongly suspect that that my fish died from a sudden dramatic pH crash. My fault through overstocking and not doing enough water changes.
Fish do go white and fuzzy quite quickly after death if you don't find them. But finding them isn't always easy. I can only count 10 ember tetras and there should be 11. I didn't find it during the last water change but with shrimps, snails and loaches, the body of a fish as small as an ember tetra probably wouldn't last long enough to get fungused. And the reason I think it has died is my fault again. I sucked an ember tetra up at the previous water change. It looked OK when I removed it from the bucket, but it could have had internal damage. There are some fish that find the siphon tube irrestistible and it's always a problem trying to avoid them.
It is easy for things to go wrong quickly in an environment like a fish tank. We've all had problems with our tanks especially at the beginning, and fish do die. Don't beat yourself up over this, learn from it.