Cases survive for all sorts of instruments, some pretty early. As some woodwinds like krumhorns, recorders and shawms were played in consorts, they were also commonly bought in sets and kept in their case. Sets of such instruments were a good addition to any fine Renaissance house, like good furniture and silver.

Stringed instruments were put in cases as well, but maybe not constantly. Thomas Mace, in his Musicks Monument of 1676, gives quite detailed information on the care of the lute. Mace was a music teacher ( much of the money for his book likely came from his previous students) and part of his book was a DIY lute tutor. Mace recommends getting a case:

a good warm Case, lined with Bayes [ Baize., i.e. cloth] within, and covered with Leather without, with Lock and Key, and Hasps, will be very necessary

But he also recommends most of the time storing the lute in the student's bed. Not putting it between the sheets , as they might be too sweaty ( yes, this is a time when laundry was not a daily thing) but storing it under the blanket. He notes the danger of forgetting the lute is in the bed and rolling over on it and crushing it, but finds it still is the best way to store the instrument. Rooms at the time were not well-heated, and English unheated rooms would often be damp: close confinement in a case in a damp room would soon result in mold, and that in turn would attack the hide glue holding the instrument together....as well as the glue holding together the case. As most stringed instruments were assembled with hide glue ( and many still are) perhaps it's more likely that, in many damp climates, the safest thing to do with something like a fiddle was to hang it up on a wall, and use the case mostly as a way for safe transportation.