The term your most likely looking for is AccessPoint, and powered over the ethernet cable is called POE (power over ethernet) and yes sounds like a typical type of setup.
I would have to assume if there were 260 people actively using 2 AP then yeah everyone is going to have issues. I doubt 260 active connections are going to work very good with 2 wifi connection to work with.
When everyone tries to get on yeah its prob not very stable.
Not much you can do - other than look get some more details, do you not get an IP address, are you not resolving where your trying to go? etc..
an ipconfig /all if your running windows from a cmd prompt will give you some info to work with.
So if you get something like 169.254 - means your dhcp did not answer.. You can view what your dhcp server IP address is and how long it gave you an IP for in the lease info. You normally try and renew the lease when its 50% of life. So I would look to see this info when your connection is working, and then again when its not?
Did you not get an IP address, can you ping your default gateway listed in the above info? Can you resolve stuff, for example does www.google.com return an IP address
My gateway in the above is 192.168.1.253 -- while your working, see if your gateway responds to ping. When it stops working, can you not ping it?
If you have an address and can ping your gateway, can you resolve www.google.com for example - depending on what this info shows you - you may be able to zero in on the actual issue.
If dhcp is the problem, you might want to make sure you renew your address just before peak usage time starts, etc. To maintain your IP during the peak, maybe your lease expires and your not able to renew during peak usage, etc.
But without details yeah its just common to have issues with an over used connection.