Occasionally, you will get a pencil where the core keeps constantly breaking. First, try changing your sharpener. If that doesn’t work, you can try gently baking them at a low heat to see if that will melt their core back together.
Awhile ago, I noticed that one my of Prismacolor Premier and Blick Studio pencil leads was constantly breaking. Utrecht sent me these instructions:
Sanford, the manufacturer of Prismacolor, has suggested that customers can put the colored pencils on an aluminum foil lined baking sheet and bake in their home oven. You would bake the pencils at 250 degrees Fahrenheit for 3 to 10 minutes. After that time turn the oven off and let them cool down slowly in the oven. Once cool try sharpening. We have heard from customers that sometimes this process needs to be done several times to reform the entire core of the pencil
I have since done this with a Prismacolor pencil, Blick studio, and Luminence pencil, and it worked brilliantly. In this case, it was my fault for dropping the pencils over and over. If you receive a set of colored pencils such as Prismacolors Premiers where all the cores are broken, then you should ask for a replacement set. They can be dropped in the store or damaged during shipping.
Please do not put your pencils in the Microwave! They could explode. No need to, when the oven trick works well. At least it did for me, your results may vary. I take no responsibility if this doesn’t work for you, try at your own risk. And please be safe!