The first time this happens, the programmer will know about 1 or two of those places, and insert difficult to find bugs all over your software.
The first time this happens the programmer will spend hours looking for the first few places that need to be changed. Then he will spend days debugging, and hopefully will find the last few places.
Then, even though he is now informed about the connections, he will NOT refactor this code because he (a) has no time to refactor as he already spent a week on what seemed like a one-line fix, and (b) is now terrified to touch this code any more than absolutely necessary.
Then the cycle will repeat.