We have found the solution. Our ODBC driver was out of date. When using 10.4 or 10.5, the you must have ODBC Driver 13. We had ODBC Driver 11. By downloading version 17.1 of SSMS (the latest version), the ODBC driver was updated and in turn, it allowed me to make a database connection to our SQL databases through ArcMap/Catalog.
We have an SQL Server (2014) with a bunch of databases on it, created by the previous data analyst. He set up this entire system and I don't know a whole lot about how it's configured. The problem is when I try to make a Database Connection in ArcMap/Catalog, I get the error message "Failed to connect to the specified server. Failure to access the DBMS server." Sometimes I also get "Unable to connect to database server to retrieve database list; please verify your server anme, user name, and password info and try again. Failure to access the DBMS server." I have tried it using both my windows authentication as well as the database authentication using the usernames/passwords left behind. None of it works. The weird thing is that I can connect to this server with no problems at all using SQL Server Management Studio, it's just a problem when I try to connect to these databases through ArcMap or Catalog.
We've tried everything we can think of, specifying port numbers (1443 [the port of our SQL server], 6080, 6443), making sure my IP is allowed through the firewall that's setup, and re-registering my machine with the IT division around here so they're not blocking anything. I can see no reason I can't connect to this instance.
We also tried enabling SQL browser services and making sure that is running. Doesn't seem to make a difference. Remote connections are allowed to the server. Additionally, my co-worker can make a database connection from his machine successfully using either authentication type (windows authentication with his credentials), but he cannot make the same connection on my machine using his credentials or the database usernames/passwords. Anybody have any ideas? I'm a real beginner when it comes to database connections and servers, so if you suggest I try something, please be detailed!