The answer to that question is "Yes".

It's a combination of the two. The receptionist and a few others have DT buttons, others just have call buttons.
As it turns out, this was not a system issue at all, despite the service providers' insistence that it was. We were able to reproduce the issue with a test set connected directly to the CO lines, bypassing the system.
We still almost missed it, because dialing each line directly worked fine. What we found, however, is that calls are typically routed through a virtual number that forwards to the lead number on the CO lines. When dialing the virtual number instead of dialing the CO lines directly, the problem occurred on a large percentage of the calls.
We were finally able to convince the provider to take a look at it and they made a change to how the calls were being routed. That appears to have fixed it.