Numerically Misleading Results

Question

Are there any calculations or results that have similar answers and when compared numerically look the same, but in actual fact after so much precision, the answers diverge from each other?

An example I could think of would be a fractional approximation for $\pi$ such as $\frac{245850922}{78256779}$. If one was unaware of $\pi$'s irrationality then they might think (through numerical "proof") that they were the same thing. Obviously this is a trivial and silly example so I am looking for examples that are more major and may even have been of a surprise to some.

Answer 1

You might enjoy the answers to this question over on MathOverflow that catalogs some examples of the phenomenon of "eventual counterxamples," particularly the Borwein integrals.

Answer 2

Ramanujan gave the identity

$$e^{\pi \sqrt{163}} = 262537412640768743.99999999999925007...$$

Martin Gardner once claimed as a hoax that $e^{\pi \sqrt{163}}$ is an integer, which apparently confused a lot of people: when they tried to disprove the claim on their calculators, they found that $e^{\pi \sqrt{163}}$ was indeed an integer - but only because their calculators were not precise enough!