The case is a special item with magical properties that most of the Rozen Maidens has. It is revealed in the start of episode 7 of Zuruckspulen that a Rozen Maiden can only rest properly in her case, and that she will lose consciousness and energy if she opts to sleep outside of it. It is also possible for a doll to use another's case when the other is awake, although two probably can't sleep in one at the same time.
Is it also possible for the dolls to use their case as a mode of transport. Suiseiseki is notable for her habit of encasing herself in her case and then using it to smash through Jun's window as a surprise entry.
A fleeing Maiden would be in deep trouble if she is unable to bring her case with her, since it prevents her from regenerating energy, and would only be able to sustain long-term activity if she used one of her sister's cases.
While the first six dolls have cases, it is unclear if the deviant dolls Barasuishou or Kirakishou have cases. They are not clearly seen with them (much like they aren't seen with artificial spirits.

References & Citations

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