Yes, but you have to be careful or you can make the person feel like they are being interrogated. Maybe not quite to the level of shining a bright light in their face and asking "where were you at 8:00pm on the night of January 23rd", but still, an endless stream of questions - done poorly - can make people feel uncomfortable even if it is true that "people love talking about themselves".
You may laugh at this, but some of the best advice I've heard on this subject came from Kezia Noble, a lady who teaches "pickup artist" material. Except... she bills herself as part of that world, but in reality her material is just general conversation skills which can be applied in pretty much any context.
I don't feel like trying to type up a complete rehash / summary of her technique here, but a big part of it involves distinguishing between "high value" answers to questions, and "low value" answers, and figuring out how to branch off from the high value answers in a way that keeps the other person engaged. There's also an element about making statements instead of just asking questions, etc.
Just to illustrate, here's a really made up example of how this might go:
You: "So, where did you go to school?"
Them: "Berkeley" (reasonably high value answer, but what to do with it??)
You: "Oh. So... what did you major in?"
Them: "Economics." (again, good answer, but can you do something with it?)
You: "Oh. Ah. Cool. So you must like math, huh?"
Them: "no". ("yes", "no", etc. are usually low value answers that don't lead naturally to anything else)
You: "Oh. Right. SO, um... do you like fishing?"
Them: "What? No. What is this, an interview?"
---------
On the other hand, here's how it could go:
You: "So, where did you go to school?"
Them: "Berkeley"
You: "Nice. Why did you pick Berkeley?"
Them: "Well, it has a reputation for great academics and a tradition of activism."
You: "Oh, yeah. My school ($WHATEVER) had a great academics as well, but not much activism." (statement instead of a question)
Them: "Activism is really important to me, especially around climate change. I wouldn't have gone to a school that didn't have that aspect."
You: "Cool. So what did you major in?"
Them: "Economics."
You: "Ah... so what's the connection between economics and climate change?"
--------
Anyway, you can kind of see the pattern. Elicit more complex responses, make statements, lead off of their complex answers, don't just suddenly branch off to something unrelated, etc.