The cappa magna is a sign of jurisdiction, and only prelates with jurisdiction can wear it, and then only in the territory of their jurisdiction. The bishop of your diocese could wear it in your diocese, but not outside of it. An archbishop can wear the cappa in his own archdiocese and also anywhere in his metropolitan province. An abbot could only wear his in the territory of his abbey.
Cardinals can wear the cappa anywhere except (I think) in Rome. They do wear it there also, but I believe there are some restrictions on its use when in the city itself.
The Pope also has a cappa, but it's use was limited even before Vatican II. I think he only wore it at Matins of Christmas, at Requiems, and at Tenebrae of Holy Week, per the rubrics. The Pope's cappa is red, not white.
The SSPX Bishops do not wear the cappa because they do not have ordinary jurisdiction anywhere. Thus also why Archbishop Lefebvre did not wear it. He would have worn it in Dakar, however, when he was Apostlolic Delegate there, and in the Diocese of Tulle, France, when he was the Bishop there.