No matter if you are using DHCP or Static addressing, so long as each of the devices you want to communicate between are on the same network, on layer 1, 2, and 3 of the OSI model, it should work. Layer 1 is the physical layer. Layer 2 is the switching layer, working with MAC addressing. Layer 3 is the TCP/IP layer, working with IP addresses. In your case, so long as each of the devices are connected via a switch, you should be fine on layer 1 and layer 2, unless your wiring or switch is having issues. For layer 3, your devices network interface cards must be bound with an IP address, whether DHCP or Static, and the network must be defined on each as well if they want to communicate with each other.
Here is an example of what you are most likely working with:
They are both connected to a switch. Computer has an IP of 192.168.1.100 and the server has an IP of 192.168.1.101.
Given the above is correct, or at least similar(The device can have an IP of anything between 192.168.1.1-192.168.1.254), then they both need to have a subnet mask of 255.255.255.0, which will define the network as being 192.168.1.0/24, meaning anything with an IP of 192.168.1.0 - 192.168.1.255 should be able to communicate with each other without needing a gateway, or as most people know it, a router. Granted, the IP 192.168.1.0 and 192.168.1.255 are generally both reserved as the network address and broadcast address, respectively, and are not used to address devices in use on the network. For more information on subnet masks and other IP information, please refer to this article.
Without going into too much detail, so long as the computer and server have an IP address within the same network and the same subnet mask, and they're both connected to the switch, you should be able to communicate between them just fine. If you can't, then there is a software, hardware, or configuration issue with at least one of your devices. Most likely, however, one of the devices doesn't have an IP address or it they both aren't on the same network. To check this, you could run an "ipconfig" in a command prompt, on each of these devices, to check what the Ethernet adapter's IP address is, if it has one. You could also check what the subnet mask is on both of them by running an "ipconfig /all" I'm assuming you're using Windows devices, but if you're not, I've listed a Linux resource below.
A good test would be to set both of them up Statically on the same network and see what happens. If it still doesn't work, you could even just run a cable between the two devices and if that fixes it, you most likely are having a switch issue.
Here's a good Technet article describing how to configure TCP/IP networking on Windows devices, as that is what I'm assuming you are using. If you are using a Linux/Unix server, then there are other great resources out there, here is one: nixCraft: Linux - Configuring Network Interfaces.
Some other great networking resources:
Wikipedia: IP Address
Cisco: Basic Networking - Switches vs. Routers
DHCP and DORA Process
TCP/IP and Subnet Masking
Switches and Layer 2
Are you sure you dont have really short lease times? – Mattias Åslund – 2013-11-08T17:27:20.963
What is your IP addressing scheme? IPs of the server and workstations? Subnet masks? – Josh – 2013-11-08T17:31:00.450
Are you trying to access the local server by IP directly or by hostname? – heavyd – 2013-11-08T17:33:58.070
Your router is clearly set to be the DHCP server. You can avoid this behavior by setting static ip addresses on each device and only allow the router handle DHCP for a different subnet. – Ramhound – 2013-11-08T17:45:07.107
there's no addressing scheme. I've since fixed two of the IP addresses. I think the staff are using the hostname. I've thought of recommending having one of the servers also act as a dhcp server and turning off the dhcp on the router. – Recognizer – 2013-11-08T17:52:58.030
Also, the lease time is set to 72 hours. – Recognizer – 2013-11-08T17:56:44.930
If they are using the hostname, where is the DNS server? – daxlerod – 2013-11-08T18:37:45.013
no dns server. Just WINS/Netbios. Join the workgroup and connect to the server kind of setup. – Recognizer – 2013-11-08T19:00:00.177