I agree with Marian that some laces just disintegrate - most '30s laces are very fragile and often too far gone to repair properly. A lot of nylon '50s-60s laces are very fragile too. I've spent many hours repairing lace dresses only to find that the more I mend, the more holes I find so I try not to embark on major restorations now unless it's really special.
I use a fine thread the same colour and gently stitch the pieces together. It's all about how strong the lace is - and different laces require different approaches. It's a delicate fabric, almost an absence of fabric but if they haven't been stored or cleaned properly it's an uphill battle. If you were in Melbourne, I could do some tutorials on how I do it but it's hard to describe online, especially without images.