Jan,
In the 35+ years I've been weaving, I've seen this happen MANY times, that's why I always try to convince the customer to use something other than flat reed splint. This breaking has only happened on two items I did, but several customers have brought chairs to me that also used flat reed splint that were broken.
Maybe suggest to your customers that they use ash, oak, hickory bark or wide binding cane instead if it's the twill design they are after. And if it's more a matter of low cost and they don't care about the design, then suggest paper fibre rush, seagrass or even Shaker tape.
I rarely if ever suggest or even for that matter, weave a chair seat or footstool in flat reed splint for this very reason. And I've taken all the precautions of using oil or actual old fashioned varnish, made sure the dowels were round and not flat slats, gave the seats good care (one was my mother-in-law's and we were living with her!), stuff the seat with foam like Shaker tape, and made sure the reed was in good condition before weaving the seats.
So, in my opinion, just don't use flat reed splint to weave chair seats with, if you do, you are asking for problems!
The Wicker Woman®--Cathryn Peters
http://www.WickerWoman.com/seatweaving-faq