There are a few things that should not be done within call center:
1. Call Forward - forwarding your phone to anything but the voicemail DN will log the agent out or make the agent not ready.
2. DND - automatically logs the agent out or makes the agent not ready.
3. Agent telephones can't be members of any hunt groups.
There isn't anything specifically mentioned about not using pickup, but it does mention that if you use pickup to get a call, you can't enter an activity code. That being said, using the pickup button defeats the purpose of having ACD.
It's hard to say what might be causing the issue without seeing the system, but it's possible one of the above mentioned items could be causing some problems.