If you have a hole in your fabric where only one or two threads are broken, try pulling a thread from the edge of your fabric and weaving it in place. Use a blunt needle one or two sizes larger than your stitching needle. You will first have to remove the broken thread(s).

I suggest starting the weave at the hole site so that you can be sure you are weaving over and under correctly, just opposite from the threads on either side of the hole so that the weave forms correctly.

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This from Teresa:

Here's a possible solution to several broken threads in linen:

If the hole is too big to re-weave a thread from the ground fabric over the hole, cut a small piece of the same type of linen, place on back of piece you are working on and when you stitch, stitch through both pieces for several stitches next to and around the broken thread. When framing I would suggest a slight padding in back to keep a lump from showing where the linen was mended.