Why can an InfoNCE mutual-information bound saturate?

Suppose an InfoNCE objective uses one positive pair and $N-1$ negative candidates. Why can its standard mutual-information lower-bound interpretation become loose when the true mutual information is much larger than $\log N$ ?

The mutual information of continuous representations is itself bounded above by $\log N$

InfoNCE becomes an upper bound once the critic separates every positive pair

The lower bound becomes independent of the critic after one negative example is classified correctly

For fixed $N$ , the conventional bound cannot certify a value larger than approximately $\log N$