Networks must be up. Slow guys have no place in this business! You need to be able to identify the problem quickly, analyze it, plan the solution and implement it. The sooner, the better. Asking you questions about meeting deadlines, or about finishing projects in time, or about being under pressure, HR managers try to find out if you can handle the pressure and work quickly.
Please notice it is a behavioral question. They ask you about a situation from the past. So you should not talk about what you would do when meeting a tight deadline, but about something that already happened. On the other hand, they do not limit it on network engineering. You can speak pretty much about meeting any deadline in any field….
Attitude is important, not the situation
Speaking honestly, one can talk about anything in an interview. You can talk about real situation, you can think up one, or you can tell them a completely unrealistic story… What matters for the employer is the way you approached the situation, your attitude. You need to present yourself as someone who can work under pressure, who does his very best to complete the work as fast as possible, because he understands the importance of networks being up and safe.
When studying at the college, I had just 24 hours to prepare a presentation from a difficult network security subject. I created a plan of work and calculated that to study the topic, prepare the first version, make corrections, prepare the presentation and practice it, I would need 18 hours. It was close to impossible to do it in time… However, I engaged my school mate to the research part, so we worked on it together and instead of 10 hours, it took us only six hours to complete the research stage. This helped me and finally I was able to deliver a high quality presentation on a next day. I managed the situation pretty well.
Once I had to distantly troubleshoot a network in a banking institution. It was during the working hours and the security modules were not working as supposed. There was a problem with both firewalls at the same time. As you can imagine, it was extremely important to deal with it quickly. I did not only start to work on it myself, but I also immediately engaged one colleague to travel there and work on the problem at place. Finally we managed to solve the problem in less than two hours.