The statement is either misleading or wrong, depending on interpretation. I defer to this answer for details, and for short version I cite my comments from under this answer
What I'm saying is, it's the right idea used in the wrong way. Yes, strictly speaking, bandlimited implies infinite knowledge: infinite sampling rate for all time. At the same time, even if we had a physical process that is truly periodic for as long as it exists, it still doesn't qualify, because it hasn't existed for infinite time in the past, and is yet to exist for infinite time in the future. Yet, this process qualifies perfectly for the motivation of the Fourier transform as a periodicity-mapping tool, and this motivation is lost when we try to be "strictly correct"
I insist on this point because of the misleading implication that, essentially, a periodic physical process can't exist. I'm no quantum mechanic but I think EM waves in vacuum defy this. The full model of "frequency" then consists of at least two relations, one being the laws that govern the photon, another the Fourier transform, and using latter without former is simply wrong. A "system of equations" if you will. If we know something is physically periodic, we should be able to call it bandlimited.
Moreover, to comment on the quote in question, the only thing that Fourier theory says about "not bandlimited" is, that sinc interpolation won't recover it perfectly. That doesn't mean nothing else will. For example, if we make a synthetic, perfectly bandlimited image, we can do full recovery perfectly. But if we simply crop the same image, it's aliasing, and we can no longer do so. I doubt anyone would otherwise call cropping "aliasing".