In my experience miracle-Gro potting soil spells a recipe for fungus gnats. If you can, get a sterile potting soil or a soilless mix. Next I'd just cut off the dead plant right to the soil level and don't worry about the dead roots. Usually these plants are crowded together and the strongest ones win out. Tease the root ball slightly and pot it in the new pot.

I've noticed with mine, since it is also in a clay pot it dries out faster than a plastic pot (which is expected) so just keep an eye on the soil moisture. Finally, a little secret I do with mine is add just a couple of drops of vinegar to about half a litre of water every couple of months just to keep the soil slightly acidic.

Oh, sorry to jump in - but I actually have almost the same question I was going to post today. But my NIP that I got for Christmas is about 4 ft tall - it is big and beautiful, but has about 4-5 stalks of which 2 are dying as I noticed yesterday. So I also need to remove these stalks somehow. I wasn't planning on repotting yet (I don't have such a large pot right now), but just wanted to remove these dying stalks. Should I try to gently pull them out with roots or just cut at the bottom?

Try pulling firm but gently, if they don't budge just cut them at the soil level.

For Viemar, I would just cut the dead stems off at the soil line -- trying to pull it out will just be messy, and quite unnecessary.

Thank you so much folks who answered my question. I had to use a saw to cut those dying trunks! They were the biggest, but not the tallest. After I cut them I discovered about 5 more small shoots popping up. Oh boy, that plant was dense! Now it looks so much better.

I have a NIP and I mist it everyday in the morning. If you do that, their branches won't dry out and die. I got mine in 2012 and it's grown a couple inches. They do well in humid environments.

It's nice that you mist your NIP every morning, but it benefits more from your good thoughts and feelings than it does from the "humidity." It's a common myth that misting a couple of times a day increases the humidity. If you had the instruments to actually measure humidity, you would see that misting raises the humidity in the air around the plant by only a few percentage points, and that for only a few minutes. Your plant is healthy because it gets enough light, and enough moisture and air in the soil, and enough but not too much fertilizer.

Misting is good in one way though -- it will show the first signs of webbing from spider mites, which are a main enemy of NIP. If you see that webbing, you can add a teaspoon of mild dish soap to the misting water, and spray that weekly, especially on the undersides of the branches, for 4 weeks. That should take care of the mites.