DHCP relies on a broadcast protocol, and like a canary in a coalmine, if DHCP stops working, then something is fouled up on the LAN.
The first things I would look at:
Has any user installed something like Spotify, Steam, Skype?
Has any user done anything brilliant like plugged in their home WLAN router with DHCP enabled into the network? A secondary DHCP server does not play well with another one on the network.
Any signs of malware infection?
Goto the command prompt on each PC and run the netstat command.
If one machine has 50 connections going to IP addresses in lower-ubekistan, there's your problem....some sort of virus. Malware will kill your network.
Long term, get a simple Ethernet hub and connect that in line with the server. Load up the free WireShark protocol analyzer on a laptop and run it. WireShark will show you what is wrong.
You can run wireshark on an individual PC and it may show you what is wrong, but it can only see traffic designated for that individual PC. It's much cleaner to run it on a laptop and just 'tap into' each switch port that you need to look at.