Oh yes! Establishing a connection between some class of natural phenomena and a well known
field of mathematics can make you famous. And you do not have to prove new theorems.
The most striking recent example is Benoit Mandelbrot.
According to the Google Scholar he is THE MOST cited mathematician of all (at the time I write this).
And all his activity was exactly as you describe.
Even before fractals, he was looking for new connections between "well known" fields
of mathematics and real world. For example he was looking for "stable probability
distributions" everywhere, "power laws" etc.
But his greatest success was "fractals". The relevant mathematics was known for about
50 years. Well known to a very narrow circle of specialists, as it happens to most
areas of pure mathematics. He invented a catchy word "fractal" and then showed by examples
that "fractals are everywhere". I don't know a single new theorem that Mandelbrot proved.
But his influence on mathematics and science was really enormous. 

On a smaller scale we have "Kramers Kronig" relations (see wikipedia).
Which is nothing else but the "well known" Sokhotskii-Plemelj formula.
It is not important that Kramers and Kronig discovered these known relations independently.
What is important is that they proposed a physical interpretation. And there are
thousands of examples like this. You can even receive a Nobel prize by establishing
a new relation to the real world of some well known mathematics.