Let $G$ be a connected, reductive group over a local field $k$, and let $^LG$ be the Langlands dual group. As explained by Borel in his article in the Corvallis proceedings, the general local Langlands correspondence should give (1) a partition of the classes of irreducible admissible representations of $G(k)$ into finite sets, called L-packets, and (2) a bijection between the L-packets and the equivalence classes of admissible homomorphisms of the Weil-Deligne group $W_k'$ into $^LG$.

When $G = \operatorname{GL}_n$, the L-packets are just singleton sets. I believe that only the local Langlands conjectures for $\operatorname{GL}_2$ were proved at the time Borel's article was written. There were no worked out examples of L-packets with more than one element at the time, as far as I know.

Why did Borel and others in the 1970s expect the L-packets to be finite? Why do we still expect this today?