It isn't with Quarter Horses and some TBs are bred with QHs sometimes. I wonder if .... nah. If they can't do it, they can't do it.
They can mix breed like that for Quarter Horse Racing. I am not sure if the Sire or Mare can be a TB, but I have seen plenty of Quarter racing horses with their grandsire or Damsire as at TB. Thoroughbreds have to be plain thoroughbreds. No other breeds in their pedigree.