When did Norby, of the series of the same name by Isaac and Janet Asimov, gain his hyperspace and time travel capabilities? My guess is he gained them when he was first made. I first put this on the Literature SE, but I figured that I might get a better answer here.
The original robot created on the dragon planet, Jamya, (by Mentor First, the lead robot left behind on the planet by the Others) had hyperspatial travel abilities. The Others had hyperspace technology and left it behind in many places, sometimes intentionally, sometimes not. Mentor First created the robot "Seeker" to explore using hyperspace travel. However, Seeker did not initially have time travel abilities (or, at least, was not intended to).
Later, the damaged Seeker was repaired with parts from an Earth-made robot by the old spacer Moses MacGillicuddy. It is conjectured by the characters in Norby's Other Secret (the "other secret" from the title being his time travel capability) that it was the combination of Seeker and the other robot (to make the Norby that Jeff Wells bought) that ultimately produced the unexpected time travel ability.