Old stems will not flower a second time, but DON'T remove them, at least not yet. Your plant needs them. A strong plant consists of minimum of three mature stems. The best time to remove surplus stems--typically the oldest stems that are leafless and yellowing--is when you are repotting. Once the plant is unpotted you can easily cut the horizontal rhizome that connects the upright stems and remove the portion with shriveled stems.
Notice that the Dendrobium nobile flowers are not carried on a single tall spike. Instead they are produced on short spikes of two to four flowers that appear at each node along the stem. It's easier to see the short spikes after the flowers have fallen (below).
Dendrobium nobile and its hybrids are a bit trickier to grow than their cousins, the Dendrobium phalaenopsis hybrids--but they are SO worth it! Check here for How to Grow Dendrobium nobile.
Wondering where to cut your Phalaenopsis spike? Find out here.
click here to find out where to cut your Cattleya spike.
To find out where to cut your Oncidium spike, click here.