I can guarantee some bright Apple Reseller salesperson has sold a BGFL school a Mac Mini Server to manage the iPads and turned on caching server as well without setting up any static public I.P.'s for the school. Thats why you are getting intermittent downloads as BGFL will be using a pool of public I.P.'s. here's a more descriptive reason..
Most LEAs tend to have a round robin type installation of proxy which means that they can have multiple external addresses. How would this affect the caching service within a private network.
I assume that the issue would be something like:-
Public IP address pool:
184.108.40.206, 220.127.116.11, 18.104.22.168
School 1 Network School 2 Network School 3 Network
192.168.1.x 172.19.10.x 10.13.12.x
School 1 places a caching service on its network at 192.168.1.10. It goes online and gets the public IP 22.214.171.124. It registers this IP address with Apple’s Servers.
School 2 has 10 iPads that go online. 5 use 126.96.36.199 and are told that a caching service exists on their network and to use it. However they cannot connect since it is a different network entirely.
The other 5 are OK since no caching service exists on any other public IP at this time.
But if the caching service was granted a different public IP the next time it went online then the caching service would update Apple with the new IP and then the affected devices would differ.