On my initial install for the management server, I also installed the
agent on it, as well as one other server for testing. Once those
started working and I was receiving traps from each one, I installed the
agent on several other servers, but I am not receiving any traps from
those servers.

One clue is that when I go into Trap Configuration and I add those
particular servers/agents, when I click Show Status, I get "The agent
<ip address> is not responding".

These servers are no different then the ones I tested with. All have
identical configurations with regards to SNMP. I've verified that
nma5.ncf is running on these servers, even unloaded and reloaded. I've
reinstalled the agent as well. No dice...