I am not a research-level mathematician, but this absolutely happens to me. There is so much in math that we can think more deeply about.
Just last week, a student came into my office to get help on a derivative problem that used the quotient 'rule'. He was getting zero in the numerator. I showed him his mistake, and said "You won't get zero in the numerator." He said he thought it could happen on some problems. I said "Well, let's prove it. Here's my conjecture: If y = f(x)/g(x), where f(x) does not equal g(x), and neither is a constant, then y' will not equal zero."
Embarrassingly, I was stuck then. I was trying to think of it algebraically, and was getting nowhere. The way to think about it (which I realized while discussing it with another teacher) is conceptually. If y is not a constant, then y' cannot equal zero. That's all there is to it. Is my understanding deeper now? My understanding of derivative was fine, but my approach to proof was too constrained. I hope I've learned something.
Pam Sorooshian, an unschooling advocate, has said that, "Once people memorize a technique or a 'fact' they have the feeling that they 'know it' and they stop questioning it or wondering about it." I think this probably happened to even the best mathematicians among us, when we were young. How much of elementary math do you take for granted? Can you get a deeper understanding of it, and still stay connected to a child's perspective? What bits of math do you 'know' to be true even though you've never personally proved them?
For years in my algebra courses, I had students using the Pythagorean theorem. It finally occurred to me that I didn't really know why it was true. (And I only knew that it was true because of all the history that references it, not because of any logical necessity.) This was not one of the times that a student question provoked me to dig deeper, but it could have been. Ever since then, I always get my students to work through the most visual proof I know.
I am not remembering all of the student questions that have made me think, but I know it has happened repeatedly. It's always a delight to me when I have to say that I don't know and I'll think about it. I never would have imagined that I could keep learning more about basic algebra, just about every time I teach it. Calculus too. And that is part of the joy of teaching for me.
I hope others (with better memories) will offer more questions students have asked, that got them thinking more deeply or more carefully.