One should surely also mention the Riemann hypothesis since Riemann computed a few zeros of the zeta function by hand and observed that they were on the critical line. I guess that this contributed to his statement, together with the functional equation, of course. (I hope that I did not overlook this response in the many previous answers, sorry otherwise).
Another experimental result with a happy ending is the birth of random matrix theory by Wigner who guessed the circular law after computer experimenntsexperiments if I am not mistaken.
Both subjects have been recently linked experimentally.
