# Why is there only a polynomial number of provers in multi-prover interactive protocols?

The paper On The Power of Multi-prover Interactive Protocols by Fortnow, Rompel, Sipser states the following:

1. There are provers $P_1, P_2, \dots, P_k$ in a multi-prover interactive proof system such that $k$ is polynomial in the size of the input. The value of $k$ cannot exceed a polynomial because this would prevent the verifier in the system from accessing all the provers.
2. In one round a verifier can interact with more than one prover.

Since a verifier can interact with more than one prover in one round, is the real reason that the number of provers must be polynomial due to the fact that the total length of the messages to all the provers in one round must be polynomial in the length of the input?