Is the spool thread catching on something (like the little notch on the thread spool) or is wrapped around a thread guide? That will tend to break the thread.
Check that you are using the correct kind of needle to match the thread and fabric you are sewing. (Are you using good quality needles? Might try a different brand.)
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All About Needles
Something else to consider is whether the thread is old or bargain bin variety. Try a different brand (preferably a better quality brand) of cotton thread. Threads have really improved over the past few years. If the thread is shredding at the needle eye or you see little fuzzies on the thread (similar to the old cotton-wrapped polyester threads of the 70's), it is not good quality and should be thrown away or used somewhere else.
Another thing to check is whether there is a rough spot on the needle plate or in the bobbin area that may be snagging the thread. Wrap your index finger in some very fine fabric (old pair of hosiery, chiffon, etc.) and rub your finger around and see if it snags anywhere.
Verify that the bobbin is loaded properly in the bobbin case.
The tension could be too tight. Remove the upper thread from the machine.
RAISE the presser foot and rethread from the beginning--confirm it is threaded correctly. Set the tension to the midway point between high & low numbers. Retest. The tension is correct when the upper and bobbin threads meet in the middle of the fabric.
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