What are these bumps on my Sterling Silver plant? They are near the end of some very long stems that are about 10+ feet long.

They're roots!

CalPolygardener said: They're roots!

Any idea why I'd suddenly be getting roots on stems that are so long?

Has it grown into a brighter light location or other environmental condition that would make it more vigorous? New fertilizing program? Basically it has gotten 'happier' and can expend energy growing roots on the stem. A lot of times this happens when the stems start to grow up instead of horizontal or down. It shows the plant is doing well.
Post some pictures of the whole plant and pot along with where it is.

Scindapsis are aroids. Most aroids make adventitious roots. This is a climbing aroid, and it is also considered a shingling aroid. It will climb a pole, or a wall, or anything it can attach to, using those roots to attach to it. The idea of growing scindapsus as a containerized hanging basket plant or a pot plant is actually not the way they grow naturally. They climb. Like this

The first pic shows how I originally had her hanging across the curtain rods. Recently she had a case of mealy bugs so I carefully went through each stem swabbing the bugs with alcohol and now she's hanging as shown in the second two pics. You can see how very long she's gotten. As I was swabbing the bugs I noticed the bumps about a foot from the end of a few of the very long stems. I thought they were related to the bugs. I'm just curious why she never tried to attach herself to the curtain rods all this time.

They need more of a flat surface. Most climbing aroids are positive phototrophs. They climb in the forest to reach more light