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 experimennts if I am not mistaken. 

Both subjects have been recently linked experimentally.