EDIT: It is possible for all characters to randomly come across a psionic institute and receive testing and possible training. When rolling for the event that happens during a specific term, the career event table can direct you to the Life Event table, which has the possibility of encountering a psionic institute. It would be rare, however. You would need to roll a 7 on the 2D Career Event roll, then a 12 of the 2D Life Event roll, then a 1 on the 1D to determine what unusual event it is. So chances of randomly encountering a possibility of testing/training would be rare (Something like a 0.077% chance on any single term if I calculated properly).