Is there an example of a “robust, yet not private” SMC protocol? I have been trying to understand the interplay between robustness and privacy of an SMC protocol. It is easy to come up with a protocol that is private but not robust. 

Can someone provide an example of a secure multiparty computation protocol that is not private but is robust, under some threshold $t$ corrupt parties?
Sure.
Parties: ​ more than 1
Input: ​ each party has a secret input
Output: ​ empty string

"Protocol": ​ Each party broadcasts their secret input then outputs the empty string.

That is robust for arbitrary thresholds, but is

not private against even a single semi-honest adversary.