So I'm trying to start the Big Dig quest. Yet Bobbi No-Nose won't open the slider thing on her door to talk. Is there a specific time it opens up? Or is this a bug and how can I resolve it?

Sounds broken.

Restart from a save point & try again? Never ideal, but sometimes it works...

I thought it just 'started' for me (when she whips open the little sliding window and speaks with you), but apparently you first need to hear from a random NPC about town that Bobbi is looking for somebody for a job...

I got it from the bartender robot in the Third Rail

The quest has a lot of scripted triggers and for some reason they can just get really slow.. A lot of the time, simply patiently waiting (standing there doing nothing which seems counter intuitive.) gets things going in the right direction again. Run around all goofy and you can screw up the quest, loot someone before something happens and you can screw up the quest, enter a room before someone else does.... you get the picture. For me, the only way through the entire quest was just going really fricken slow.

I'd try a reload from before you entered the area. Like from another loading zone like from inside the bar. Slowly walk up to her door and then just stand there for a bit.