I used BFS on a weighted graph in an interview once. It gave the wrong answer. I did not even realize it was wrong until the interviewer asked me to trace the execution. That was the day I learned that BFS has a hidden assumption: all edges cost the same.
This graph has 4 nodes and 4 directed edges, each with a travel time (weight). The source is node 1. We want the MAXIMUM of all shortest distances — the last node to receive the signal. Run BFS on it and watch what happens.