We are having random issues connecting to internal servers and computers. Computers can connect fine one moment and can't connect at other times. At the same time that one computer can't connect, other computers can connect. This problem happens randomly and we can't figure it out. It is a DNS issue? Router issue? We have 1 router and a switch.

Sounds more like a license issue than a dns one.
What makes you think its dns?
what is the message you get when you can't connect?
are the workstations pointed to the ms server for dns or the isp?

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Assuming your 1 router and 1 switch are on 1 logically addressed network then it would not be an issue with the router.

Are you attempting to access these servers by their names or IP addresses? If using mapped drives, are the drives mapped to names or IP addresses?

Once we have a little more information we will be able to help.