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My first feeling is that the soil has dried out so much that water just runs right off and is not absorbed. Try submerging the pot in the sink or a bucket of water, and just let it sit there for an hour or so to thoroughly wet the soil. Then pull it out of the water, let drain thoroughly, then replace on its saucer. Check the soil moisture every few days by digging into it with a spoon, at least half the way into the pot. It should be damp enough to almost stick together when you squeeze it, but leave no visible moisture on your fingers. Soft and cool and nice is how the soil should feel, then you can water it again. You can cut off the dead curled ends from the branches, but be warned, NIPs don't like going so dry, and it might not make it. But don't be discouraged - try again if you lose this one.

If you're talking about lower branches, that would be normal. If you're talking about upper branches, the first thing that I would think of is the soil is too dry. Another common problem of NIP's indoors is spider mites.
To see if the plant has mites, take a spray bottle with plain water in it, and squirt along the branches (The only useful aspect of the common practice of "misting"). If there are mites, the water will stick to the webbing, and you will see it instantly. You can eradicate the mites by spraying with a mixture of 1 tsp of liquid dish soap in a spray bottle of water, sprayed all over till the plant is dripping, especially on the undersides of the leaves. Do this every week for a month.