There are a fixed number of lines going into and out of every phone system. Based on 25 years experience with these systems, there are only four plausible explanations: 1) As configured, the phone system is failing before all trunks are busy which is a design issue. 2) The system failed for a reason unrelated to call volume, i.e. it's a coincidence. 3) The system recording voice mails ran out of disk space. 4) They intentionally shut the incoming trunks down.
And if that turns out to be true, that’s just on more nail in their coffin come election time.
OR the volume overloaded the processor (which may be the case)