I cut off some black spots when peeling my russets, but when I boiled the potatoes a few more appeared. Why is that?

What you probably experienced is called “black spot bruising” which can occur when potatoes have been stored too cold (typically below 40°F) or the potato container (box or bag) is dropped or jostled in shipping or when put on display.

It’s hard to detect every time the potatoes are taken out of storage and sorted into consumer bags for the retail stores because, as you found, the outside of the potato may look fine. The dark areas, usually at both ends, are often not visible. As with humans, a bruise or bump can show up on or just under the surface of your skin but it could be a deeper issue. Cutting off the black spots catches most of it, but heating the potatoes brought the other parts out. The only answer at that point is to trim a little more off the potato and discard.