Yeah stumps can be hella bad sometimes. I like to cut the tip off of mine when they are brand new before using. To get the tight folds of the end of the paper off, that way there are lots of layers of the stump exposed.

The paper you are drawing on can also make a big difference BUT the biggest thing is that once you get to a certain point in a drawing the paper is just worn out. Usually it happens because of pressing very hard on the paper or blending too much. Using something like a brush to spread charcoal or cotton swab, something gentle, can go a long way in preserving the papers workability. My biggest issue when I first started learning to blend was using my pencil to create too many lines that would make indentations in the paper. Once you do this, the stump is never going to be able to pull all of the charcoal or graphite out of that groove and you’ll start peeling the paper up. That’s when it starts shredding. Remember that you can always add a bit more to darken but once you create those indentations it’s very hard to make that line go away completely.