I'll get to the solution first for this issue right off the bat. To join a Windows domain, you have to be able to connect to a domain controller using host name resolution. The simplest way to test that is working is to ping your domain name (e.g., companyname.int):
ping companyname.int
If you get a response from one of your domain controllers, you should be able to join the domain. If not, you won't be able to join the domain.
In our instance, we had a laptop connected over our wifi that was failing to join the domain. We'd been able to join laptops to the domain over wifi previously, so after some help from Google, we discovered the "ping your domain" test. That was not working.
ipconfig/all showed that the first DNS server in the list of DNS servers was 8.8.8.8. Huh, well, internal name resolution isn't going to work using that DNS server! We manually removed 8.8.8.8 from the list of DNS servers on the laptop and it was able to join the domain.
It turns out that our network admin configured 8.8.8.8 as the first DNS server on the DHCP server running on our office's main router several months back when we had some internal DNS server issues. At that time, Internet access stopped working for the office, since DNS was no longer working, and adding the 8.8.8.8 DNS server fixed that. Probably would be best to added it to the end of the DNS servers list instead.