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Perplexed comments on How to Convince Me That 2 + 2 = 3 - Less Wrong
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Arguably this is the case for everything (until we solve the problem of induction). In the meantime, I don't know of anything you can't assign a probability to or collect evidence about.
As for whether this is an analogue to Godel's theorem (or, in times gone by, Russel's paradoxical catalogues - or in times yet to come, the halting problem) - no. Mathematical systems are useful ways to carve reality at its joints. So are categories, and so is computation. They can't answer questions about themselves. But reality quite clearly can answer questions about itself.
How about the question of whether there is anything you can't assign a probability to or collect evidence about?
You can assign a probability to that. I hadn't considered the question strongly enough to have a mathematical number for you, but I would estimate there is a 10% chance that there are things which I cannot assign a probability to or collect evidence about. (Note that I assign a much lower probability to the claim "you can't assign a probability to or collect evidence about x"; empirically those statements have been made probably millions of times in history and as far as I know not a single one has been correct)
That said, "I don't know of anything you can't assign a probability to or collect evidence about" is true with a probability of 1 - 4x10^-8 (the chance I am hallucinating, or made a gross error given that I double-checked).