Are you talking about the flower stem? or the orchid plant istelf? (pseudobulb / cane / etc).

Unless the cane/p-pulb has a disease, you should leave it there until the plant discards it on its own.
It may still have some nutrients stored that the plant can use as reserves.

If it is a flower stem, then you can cut it as soon as it starts to show signs of deterioration. You can cut it about 2 inches from the origin of the spike.

the orchid has 2 main stems, on the one stem two branches are turning brown and are almost completely dried out. however the stem itself is also slowly turning yellow and it's spreading. there is still one branch left on the stem that has a flower on it and is growing another bud. the yellow on the stem is spreading quickly.

The term we use for that is spike. You can't go wrong with the saying, "If it's yellow, let it mellow. If it's brown, cut it down."