TITLE: Theorems that are essentially impossible to guess by empirical observation
QUESTION [99 upvotes]: There are many mathematical statements that, despite being supported by a massive amount of data, are currently unproven. A well-known example is the Goldbach conjecture, which has been shown to hold for all even integers up to $10^{18}$, but which is still, indeed, a conjecture.
This question asks about examples of mathematical statements of the opposite kind, that is, statements that have been proved true (thus, theorems) but that have almost no data supporting them or, in other words, that are essentially impossible to guess by empirical observation.
A first example is the Erdős–Kac theorem, which, informally, says that an appropriate normalization of the number of distinct prime factors of a positive integer converges to the standard normal distribution. However, convergence is so slow that testing it numerically is hopeless, especially because it would require to factorize many extremely large numbers.
Examples should be theorems for which a concept of "empirical observation" makes sense. Therefore, for instance, theorems dealing with uncomputable structures are (trivially) excluded.

REPLY [3 votes]: A slightly old question, but I've only just seen it...
Lord Brouncker proved in 1654 that
$$
\dfrac{1}{4}\pi
=
\cfrac{1}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\dotsb}}}}
$$
This has very slow convergence. Apparently people didn't believe it initially. It takes nearly 50 terms for five decimal places of accuracy and nearly 120 for six! It converges in an oscillatory manner and the jumps are quite large for quite a long time.
Of course, 120 terms isn't that many for modern computers and there would be no doubt about it these days. But, 120 is a lot to do by hand in the 1650s!