Suppose Alice has a bit string of length $n$ where $n/2$ bits are chosen uniformly at random to be 1's; and the rest are 0's. Alice sends a message to Bob.
If Bob needs to reconstruct the bit string, then $\Omega(n)$ communication is needed.
However, suppose Bob only needs to reconstruct at least $0.99n$ bits correctly (without knowing which bits are correct). Would this lower bound still hold?