I do self-board (as a teen and now as an adult), but follow the ups and downs of the life of my friend who is now running a boarding barn. I know my hay and feed reasonably well, and I also get to see the reactions of all the other self-board folks at my barn.
First, yes, hay does vary from bale to bale and from batch to batch.
Second, most horse owners In full board, and many horse owners in self board, do not know much about hay and feed, or how to evaluate hay. For instance, they may overfeed a horse and then say he “doesn’t like” the hay because he isn’t finishing up his meals. They may mistake dry alfalfa with the leaves breaking up for “dusty, moldy hay” (I got a real good deal taking that batch off their hands!). They may think that stemmy hay is low calorie (and give their pony founder by feeding it). They may be unable to differentiate between an obese horse and one in good condition, or to differentiate between a horse that lacks topline (because of protein or mineral deficiencies or the wrong work load) and one that is actually underweight.
Third, if you your horse is in full board you are paying the BM to take over day to day care. No place will give you the obsessive attention to detail you can accomplish in self board or on your own property. You have to decide if the overall level of care and facilities is satisfactory, and if it isn’t, then move.
The best way to evaluate your horse’s condition is to look at his condition. If he is keeping a good weight on the barn’s feed then the hay is OK. If he is losing weight, then either the hay is not OK or he is not being fed enough. If your horse looks fine but “you don’t like the hay” then that may not be a real issue.
Fourth, I would think that any BM in a well-run full board barn would not be allowing random clients to bring in random bales of hay. Where will they be stored? How will you keep them out of general use? Who will feed it?
You may be able to do this in some looser-run or backyard places, but in general, if you are paying $700 to $900 for full board at a competent place, why would you even be considering bringing in an extra $100 worth of feed a month?
So, to sum up.
Most horse owners don’t know much about hay, and if they complain about the quality, the BM is probably rolling her eyes behind their back.
On the other hand, if the feed is consistently substandard then your one option is to move.