OK to keep interviewing while background check is pending

I received an offer in writing from a BB and the background check is taking a long time (going into week three), because I lived abroad for a few years. As a result, my start date has been pushed back one week.

A tech firm contacted me last week to interview for a position that I applied to prior to receiving the offer from the BB. I debated and finally accepted the interview, because I thought that it could be a good opportunity to make a connection with someone doing similar work in a very different industry. Additionally, a few days ago a recruiter from a BB where I previously interviewed contacted me about a position. I am not sure if I should reach out to the BB recruiter to find out more about the position or if I should consider moving forward with the hiring process if I make it to the next round with the tech company.

Advice on how to proceed in this situation seems to be contradictory. I have heard that it is OK to interview up until (or even a week or two past) a start date, just in case something falls through. I have also heard that an offer can be revoked for this sort of behavior. Does anyone have any advice on how to proceed? I would feel like an idiot if something fell through with the BB and I turned down these great opportunities. However, I don't want to get caught with my hand in the cookie jar if it's something that could get me blacklisted.