From what little records exist, we know that body-snatchers required some level of moonlight in order to conduct their work in cemeteries, although not all bodies were obtained through exhumation. The clothes and burial shroud were sometimes removed, for stealing a body on its own was not considered theft since it had no value as property.
The body-snatchers might steal as many as six bodies in a single night and often worked in small gangs which fought each other for ‘a monopoly over the cadaver trade’.  This might involve desecrating a graveyard that supplied bodies to a rival gang in order to arouse fury from the local population who would then secure the cemetery, making it difficult for future attempts.
Grave robbing led to the development of devices used to protect graves. It also led to suspicions that body-snatchers also murdered people, which wasn't helped by the rare cases in which some did. And the stories fueled fears of being buried alive. One account survives in which a man was saved from such a fate by grave robbers! Read the story at The Chirurgeon's Apprentice. Link -via Not Exactly Rocket Science