I just read this blog post by Terrence Tao, in which he talks about the three stages of mathematical education---pre-rigor, rigor, and post-rigor:
It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. Unfortunately, this has the unintended consequence that "fuzzier" or "intuitive" thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as "non-rigorous". All too often, one ends up discarding one's initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one's mathematical education.
The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.
I do think this applies in other fields as well, even experimental sciences. My own personal examples include statistics and thermodynamics. For the former, I feel as though for certain aspects I've reached the stage where I'm fairly comfortable with the theory and I no longer need the absolute rigor of going through systematic symbolic manipulation to get where I'm going. On the other hand, for thermodynamics, I definitely plateaued at the stage where I was lost in a sea of symbols with no intuitive way out. I could calculate a lot of rigorous conclusions, but it felt like I'd lost my inner sense of direction. Terrence Tao's post definitely inspires hope that this is a natural progression in my march to actually understanding any subject.