It could be. For example, there could be some (as-yet undiscovered) property of the Universe that is not computable by a Turing machine (but also not useable by us to construct any more powerful computer) to arbitrarily good approximation, in which case, we might well be out of luck when it comes to describing it with mathematics, for the most part. This is not guaranteed, because non-computable doesn't necessarily mean non-describable (c.f. Chaitin's constant), but it could also be that, too, and in any case its non-computability may make it frustratingly difficult to analyze.
There is no guarantee that "reality" has to make things easy for, or even accessible to, us. Thinking otherwise is really just a form of conceit, however much layers of philosophizing you may choose to wrap it up in. Anything we can imagine to fit in a gap in our knowledge just might fit there as long as it doesn't evidentially contradict anything we've already established. It is, say, entirely possible that all elementary particles are actually small gnomes that just happen to follow to the same precision that we've verified as being accurate, all the laws we have already devised for such things. Of course, when it comes to investigating things, we have to have some guide as to what ideas will be most likely, promising, or useful for thinking about, hence if this theory doesn't tell us anything new or useful, it's probably not worth it to spend time or money on, but there is a real and important difference between saying something isn't worth it, and calling it impossible. If nothing else, it maintains an adequate sense of intellectual humility, something I find too often to be missing.
But then you may ask "doesn't this lead us to just give up?" For one, if you're going to use that as an argument for how "reality" is, then you are doing an "Appeal to consequences" fallacy (It also has some shades of Pascal's Wager). On the other hand, no - whether or not to give up is not a function of if we can or cannot understand everything or even just everything accessible to us (another important distinction), because we also cannot prove something is never understandable, either. It could be, or it could be that the next model to understand it is just over the hill, or simply extremely complicated (again, no obligation to make things easy for us). Granted, if after a very long time (which I'd put at hundreds, to thousands, of years or more), it never cracks, maybe we can start to put evidentiary confidence behind such an idea, but not before then. The trend has been clear for a while now that each unknown we've run across has since become known, and as someone who's done a quite fair share of arguing with global heating deniers, I would have to say that one should never focus on a relatively brief, seeming pause as disproving a trend. Trending quantities fluctuate - to refute the trend, you have to see a deviation over a scale that is much larger than that of the natural noisiness within the quantity the trend fits.