I have a Chico that is close to breeding limit can I draw straws to hreed after the breeding limit?

there is no way to get around the breeding limit unless you get lucky with twins on the last breeding. (which is pretty much impossible as twins are never a guarantee).
Chincotegues, Marwaris and Akhal-tekes are all limited to 5 live foals and 3 lab pulls. note that you can spend more money/ingots on the lab pulls for a chance of more eggs and straws, but I haven't found it worth the cost to do so though. It's easier to just buy new breeding stock off someone 

Thank you

But what about breeders promise?

Lop rabbit, no. If it did work, then that would be a bug, and you'd likely get in trouble for exploiting the bug. 

Umm I never had to deal with limited breedings?

The breeds Wireless mentioned have always had breeding limits. Its just those breeds. 

Oh!