A lot of time has passed, so perhaps this question is moot for you [I mean, I saw you give the Plücker lectures last week on something very different], or perhaps you now know the answer. But since the question was of interest to me for a long time, and since I didn't see the answer until a few years after it was published, here it is for the record.
Multiplicity can occur. Henniart and Vigneras give an example of a group $G$, a maximal compact subgroup $\tilde K$, and a depth-zero cuspidal representation (really, a character) $\chi$ of $\tilde K$ such that the Hecke algebra $H(G,\chi)$ is noncommutative. This implies the existence of higher multiplicities. The example is somewhat hidden by the title of the paper: A Satake isomorphism for representations modulo $p$ of reductive groups over local fields, J. reine angew. Math., 2015.
See $\S$4.4, and in particular the remark there: the example can be made to work when the coefficient field is complex.