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Mishna: If one covers his sukkah with the long boards of his bed, it is not kosher schach.
Summary
The long boards that were broken off of the side of a bed seem to fit the requirements for schach (9.3). They are wood [‘from the ground’] and are not mekabel tumah [only something that is a “keili” can be mekabel tumah, but once a keili breaks, its parts are called shivrei keilim and are no longer mekabel tumah].
Suggestion: This mishna seems to support Rabbi Ami bar Tivyumi.
Rabbi Ami bar Tivyumi: Worn out clothes are possul to use as schach.
It seems that while shivrei keilim are not mekabel tumah, there is a rabbinic prohibition against using them for schach. This is why boards of a bed are not kosher despite their being shivrei keilim.
Response: The long board of a bed is not considered shivrei keilim as explained by Rabbi Chanan quoting Rebbi elsewhere “it is referring to the long side board with two legs or a short side with two legs” [which is still useful as a bed]. Perhaps here too it is referring to the long side board with two legs or a short side with two legs.
Question: What is this statement by Rabbi Chanan quoting Rebbi?
Answer: There is a mishna:
“Rabbi Eliezer: A bed is mekabel tumah only when bundled [all parts are built]. A bed can be purified [if it became tamei] only when bundled. Chachomim: It is mekabel tumah in parts, and it can be purified in parts.
Question: What type of parts are the Chachomim referring to?
Rabbi Chanan quoting Rebbi: The long side board with two legs or a short side with two legs.
It is fit to be supported by a wall and thus used as a bed.
(Actually we pasken like Rav Ami bar Tivyumi that shivrei keilim cannot be used for schach. So our mishna need not be explained as referring to a board with legs, it can be referring to a broken piece of furniture. Still it’s possul because of a rabbinic prohibition).