This paper carries on the effort to bridging runtime verification with distributed computabil- ity, studying necessary conditions for monitoring failure prone asynchronous distributed systems. In a previous paper (to appear in RV 2014), it has been proved that there are correctness prop- erties that require a large number of opinions (e.g., true, false, perhaps, probably true, probably no, etc.) to be monitored. The main outcome of this paper is that the need for this large num- ber of opinions is not an artifact induced by the existence of artificial constructions. Instead, monitoring an important class of properties, requiring processes to produce at most k different values does requires a large number of opinions. Specifically, our main result is a proof that it is impossible to monitor k-set-agreement in an n-process system with fewer than min{2k, n} + 1 opinions. The lower bound is shown to be tight.