I re-read your post and realized the issue is the thread cutter.
Is this happening on a machine embroidery project when the machine cuts threads between stitches?
Can you pull off the needle plate and check down in the machine for any pieces of loose thread?
When you say the thread snaps, where is this happening? Is the thread breaking before going into the tension discs or at the bottom in the needle area?
Do you have a setting to use different lengths for thread cutting? Check that the foot and or holder are secure on the machine.
What type of thread are you using? How does it come off the spool? Is it metallic?
Check the machine preference pages and re-set to default settings (if this feature is on the machine.)
Try a different bobbin, or spool of thread, or even a different embroidery design (if machine embroidery).