Further information this morning:

The system CAN make 6 keys, but they have it configured to reset after 4 for "security." This is baffling to me. The real security issue is issuing keys to unauthorized people. If key 5 is issued to an unauthorized person, only the legitimate guests are locked out. I can't see how limiting rooms to 4 keys makes them more secure.

The most likely security problem is caused by refusal to issue a key to every guest in the room. In order to avoid being locked out, the keyless guest is likely to use the security latch to hold the door open/unlocked every time she needs to run down to the lobby for something, visit someone in another room, etc. It increases the odds that doors will be left open.

Additionally, when I return at 2:00 AM and try to enter through the back door nearest my parking space, but my key doesn't work because one of my roommates had to get a new key, that creates a real security issue.

If the hotel had disclosed the four key limit on its website, there's a good chance I would have booked elsewhere.