Dry hay has to probably be below about 15% moisture -- going into dry storage -- to be 'safe from storage spoilage'.  Even at 20 % moisture ( upper limit for small square bales), hay will heat in storage (minimal,  final respiration).  Expected heating at this moisture level should only reach about 110 to 120F in storage – and this is considered ‘normal heating’ or ‘the sweat’.  As long as the residual moisture and heat can dissipate, the highest storage temperatures is usually reached at about day 4 or 5 of storage then declines.  If the heat cannot be dissipated (good ventilation, or not covered tightly with plastic immediately), then temperatures above 120 F can lead to ‘greater than normal’ heating, dry matter loss, decrease in protein digestibility, and mold.    If small square bales are stored at greater than about 20 % moisture, heating goes higher and more  ‘heat damage’ should be expected.

Because large round and large rectangular bales are more dense ( and dissipate moisture and heat more slowly), the upper limit of safe baling / storage moisture  for large round bales is about 18% moisture; and for large rectangular bales, about 16 % moisture.  

If otherwise ‘safely stored’ dry hay is re-wetted, it can again re-activate the microbial organisms present and start the heating and deterioration process all over again.  This is pretty common in outside-stored bales. 

‘Dry hay’ will absorb moisture from high humidity air; pushing ‘border-line safely stored hay’ into the mold and spoilage range.  This can happen to hay produced in one region and shipped to a region where humidity is more consistently high ( Florida, coastal Texas).  

 This ‘equilibration’ with air moisture often  goes the other way in the winter.  Under very low winter humidity conditions, inside stored hay will often dry down to less than 15% moisture.