I do not think there is an exact answer to this question. My experience is that it depends totally on the donor. I have seen very big differences between different cows and I have not been able to figure out how you can tell which donors can be flushed extensively before being rebred for a natural calf. I have one donor who was flushed 9 consecutive times and averaged 27 grade 1 embryos. I then brought her home from the ET center and she was bred to a bull on pasture, 6 days after her last flush. I never thought she would conceive to that breeding, but she calved with twins in 2013. She then moved up to calve 5 weeks earlier in 2014 at the age of 12. I have had other donors that were completely messed up after two flushes. I have had 2 donors that never rebred after being flushed 2 times. After trying most everything to get them to rebreed, I eventually ended up sending them to market. I have had many more donors that I have flushed for a full year ( 6 flushes) and then rebred them for a natural calf. I also had one donor that I flushed for almost 7 years non stop and she was still pumping out great numbers of embryos when I retired her. The only reason I did this is because she got a viral mastitis that destroyed all 4 quarters when she was 5 years old. She almost died but I was able to keep her alive. For about 3 weeks, I carried feed and water to her, because she could not get up, and it was a small miracle when she eventually did. She was too thin to ship after that happened so I turned her out on grass but did not put a bull with her as she could not raise a calf. By mid summer she was fat, so since she had been an excellent brood cow, I decided to try to flush her. She generated over $70,000 in embryo sales and progeny and I have some excellent daughters in my herd from the embryos she produced. Her embryos have sold to 7 countries and I still have some in my inventory. I have attached a picture of this cow at nearly 13 years of age, when she came home from being at the ET center for almost 7 years.
In my opinion, I think the time between flushes is an important factor. We never flush more than 2 months apart. It gives the donors a few weeks to rest between flushes.